Category Archives: Physics

What exactly is a sub-atomic particles structure like?


We don’t really know yet. String Theory says that it is a twisted-up 10-dimensional space called a Calabi-Yau manifold like the one shown here (Credit:Wikipedia).

When you are asking what something looks like, you are asking to describe it in relation to its surrounding space. The problem with asking this question of a fundamental particle in the quantum worldis that we do not have a working understanding of what space[time] is like at these scales. Without THAT, we cannot stand even begin to answer this question. There are some indirect clues that could help us constrain the answer in a meaningful way.

Take the simplest particle we know about, the electron. Experiments show that it is just a knot in the electromagnetic field with no solid surface or internal structure. As you scatter more and more energetic particles off of it, all you see is a region of increasingly higher electric field strength. One thing is for sure, it cannot be a tiny sphrical ball with its electric charge on its surface. Such a system would explode instantly from the enormous electrostatic repulsive forces. Also, because they obey special relativity, they cannot have an extended shape because as they travel through space, their shape would change and no two observers would agree who was observing the correct shape for an elementry particle. Any physical extension would violate

A theory called quantum electrodynamics has been tested to a phenomenal number of decimal places, 10 or more, and still agrees with experimental data on the electron and how it behaves quantum mechanically. The theory stipulates that the electron is a pure point particle with absolutely no internal structure. If you added internal structure, the theory would violate special relativity.

Recently, experimenters have found that in certain kinds of experiments, there MAY be a weak departure from the predictions of QED and experiment, but the scale at which this happens is about 10-20 centimeters or so, and at energies above 100 GeV. Future experiments will check this.

Also, some recent results at Fermilab seem to indicate that quarks may have some internal structure making them less than fundamental, however, these experiments are far less conclusive and under considerable controversy given the statistical significance of the results.

On the theoretical side, it has been widely expected for decades now that at a scale of 10-33 centimeters, the structure of space-time will cease to be the implacable, smooth ‘surface’ we use in modern quantum theory. It will become something quite bizarre like the figure at the top of this page. Perhaps an unimaginable froth of mini-worm holes, quantum loops or strings wiggling about in some strange kind of hyperspace with a dozen or more dimensions. At these scales, all particles loose their point line character. All quantum fields reduce to some more complex topological structure as the superstring theorists advocate.

Conceptually, I have not the slightest idea how to interpret the mathematics, however, if the mathematics lead to testable and verifiable predictions, how we think of the mathematics becomes a moot issue. Just like we cannot accept quantum indeterminacy, duality, and special relativity as ‘common sense’ ideas, yet this is the way nature seems to work.

A Gallery of Geomagnetic Storms with DIY Equipment

It has been over a year since I published my book on how to build DIY magnetometers that can detect geomagnetic storms. The $8.00 B/W book ‘Exploring Space Weather with DIY Magnetometers‘, is available at Amazon by clicking [HERE]. It contains 146 pages with 116 illustrations and figures that describe six different magnetometers that you can build step-by-step for under $50.00.

Over the last year I have posted at my Astronomy Cafe blog, and at various LinkedIn groups, the magnetograms from my most sensitive magnetometers to show how well they capture the rapid changes of Earth’s magnetic field during a geomagnetic storm. Since I started posting these DIY magnetograms on Linkedin, they have received over 13,500 views so there seem to be a lot of teachers, amateur astronomers and space weather enthusiasts interested in my DIY technology.

On the NOAA Scale, these storms are stronger than G2 and are currently happening every month or so during the sunspot-maximum period. This blog is my Gallery of the storms I have detected so far. I also show the data for each storm event observed from the Fredericksburg Magnetic Observatory (FRD) located about 200 miles south of my suburban Maryland location. This will give you a sense of just how accurate my designs are compared to the far more expensive, professional-grade systems.

By the way, the July 14, 2023 magnetogram ‘spots’ shows what can be accomplished by a simple $5.00 soda bottle magnetometer if you follow a design with a laser pointer and a 7-meter projection distance as described in my book.

Ok…so here are the magnetograms in reverse chronological order starting from the most recent storms and working down the list to the earlier ones towards the end of 2023. I am only presenting the magnetograms and not a lot of supporting information about the circumstances of the storms themselves. For this information, visit the Spaceweather.com website and in the upper right corner of the webpage in the Archive area enter the date of the storm and you will be able to see a lot of info and even amateur photos of the resulting aurora themselves.

Blog 1: DIY Magnetometers for Studying Space Weather

Blog 2: The Great Storm of May 10, 2024.

Blog 3: The Minor Storm of May 13, 2024.

October 10-12, 2024, Major Kp=8-9, Great Aurora. FRD magnetic observatory plot (red) versus the RM3100 (black). Single-digit Kp index numbers on top row (from 2 to 9). The features that look like sudden ‘glitches’ at Kp=9, 8 and 7 seem to be very real and rapid changes in the D-component (angular displacement). 1000 units on the vertical axis corresponds to 200 arcseconds or 3.3-arcminutes variation.

August 12, 2024. Major Kp=8 storm. Green arrows are the Sq current variations. FRD (red) and RM3100 (black).

August 2, 2024 Kp=7 storm event. FRD(black) and RM3100 (blue).

June 23, 2024, Kp=8. Sq minima (arrows). Storm event (blue bar).

May 12, 2024, Kp=6 storm. FRD(red), RM3100 (black), Photo (blue), Hall (green).

May 10, 2024 Kp=8-9 major geomagnetic storm.

May 5, 2024 storm Kp=4-5. FRD(red), RM3100 (black)

March 23, 2024. Diurnal Sq dips (arrows) and a strong geomagnetic storm (hour 40). FRD data (red) and RM3100 data (black).

November 27, 2023. Minor Kp=6 storm at running UT of 83-86 . FRD data (yellow), RM3100 (grey), photoelectric magnetometer (orange), Hall sensor (blue)

November 5, 2023. A significant Kp=7 geomagnetic storm superposed on a few wobbles due to Sq current effects.

October 30, 2023. Three days of Sq variations and no geomagnetic storms. FRD data (red), RM3100 magnetometer (black), photoelectric magnetometer (blue), Hall effect magnetometer (orange).

July 14, 2023. Kp=4 geomagnetic storm (blue bar) with three cycles (yellow) of the diurnal Sq current. Red line = FRD data. Spots = soda bottle magnetometer.

The Minor Storm of May 13, 2024

We had a minor geomagnetic storm on Monday just after the major storm on Saturday that everyone saw. This minor storm launched a CME caused by an X-5.8 solar flare on Friday, but despite early estimates it might rival the major storm, it was a glancing blow to Earth’s magnetic field and caused no aurora over much of the Lower-48 States. Many had hoped they would get to see an aurora in Maryland and other mid-latitude locations but the storm was too week to be seen in most states that had enjoyed the Great Storm of May 10-11.

Nevertheless, my DIY magnetometers did show some life for this Kp=6 event as shown below. This time I had three different magnetometers operating. The top numbers are the 3-hour Kp indices. The red trace is from the Fredricksberg Magnetic Observatory. The black trace is from the RM3100-Arduino system. The blue trace is from the Differential Hall Sensoe system. The green trace os from the Differential Photocell Magnetometer. The two dips marked with ‘Sq’ are the diurnal Sq variations, which were recorded by all magnetometers.

All three designs are described in detail in my book Exploring Space Weather with DIY Magnetometers,

Landscape Dimming During a Total Solar Eclipse

During a solar eclipse, the lansdcape will slowly dim until it is nearly complete darkness along the path of totality. other observers wil see te landscape dim a bit but then brighten to normal intensity. If you didn;t know that an eclipse was going on you might not even notice the dimming, mistaking it for a cloud passing across the sun. The geometric condition for this dimming have to do with the area of exposed solar surface and how this changes as the disk of the moon passes across it. Below is a simple mathematical model for ambient light dimming that you can put to the test the next time a solar eclipse passes over your geographic location.

I have reanalyzed the geometry and defined it in terms of the center-to-center distance, L, between the sun and moon, and their respective radii Rs and Rm as the figure of the upper half-plane of the intersection shows, with the yellow area on the left representing the disk of the sun and the white area on the right the disk of the moon. This problem was previously considered in 2000 by British astronomer David Hughes who used the distance defined by the segment FE, which he called alpha, but L = 1+M-a. The figure shows the moon overlapping the disk of the sun in a lens-shaped zone whose upper half is represented by the area AFDE.


The basic idea is that we want to compute the area of the lunar arc cap AFD by computing the area of the sector BAF and subtracting the triangle BAE from the sector area. That leaves the area of the cap as the left-over area. We perform the same calculation for the solar sector CAE and subtract the triangle CAD from this.  The resulting area of the full lens-shaped overlap region is then

Occulting Area = 2x(AreaAFD + AreaAED).

Because of the geometry, the resulting area should only depend on the center-to-center separation and the radii of the sun and moon. You should not have to specify any angles as part of the final calculation. In the following we will use degree measure for all angles.

The area of the sector of a circle is just A = (Theta/360)piR2 so that gives us the first two relationships:

To simplify the problem, we are only interested in the fraction of the full sun disk that is illuminated. The full sun has an area of pi Rs2, so we divide Am and As by pi Rs2 , and if we define Rs=1.0 and M = Rm/Rs we get:

Although M is fixed by the solar-lunar ratio, we seem to have two angular variables alpha and theta that we also have to specify. We can reduce the number of variables because the geometry gives a relationship between these two angles because they share a common segment length given by h.

so that the EQ-1 for A can be written entirely in terms of the center-to-center distance, L,  and moon-to-sun disk ratio M = Rm/Rs. This is different than the equation used by Hughes, which uses the width of the lens (the distance between the lunar and solar limbs) segment FDE=a as the parameter, which is defined as L = 1+M-a.

During a typical total solar eclipse lasting 4 minutes, we can define L as

L = 1900 – 900*(T/240) arcseconds where T is the elapsed time from First Contact in seconds. Since L is in units of the current solar diameter (1900 asec) we have

EQ 3)          L = 1 – T/480.  

If we program EQ 1, 2, and 3 into an excel spreadsheet we get the following plot for the April 8, 2024 eclipse.

First Contact occurs at 16:40 UT and Fourth Contact occurs at 19:57 UT so the full duration is 197 minutes. During this time L varies from  -(1+M) to +(1+M). For the April 8, 2024 eclipse we have the magnitude M = 1.0566,  so  L varies from -2.0566 (t=0) to +2.0566 (t=197m). As the moon approaches the full 4-minute overlap of the solar disk between L=-0.05 and L=+0.05 (t =97m to t=102m), we reach full eclipse.

We can re-express this in terms of the landscape lighting. The human eye is sensitive to a logarithmic variation in brightness, which astronomers have developed into a ‘scale of magnitudes’. Each magnitude represents the minimum change in brightness that the human eye can discern and is equivalent to a factor change by 2.51-times. The full-disk solar brightness is equal to -26.5m, full moon illumination is  -18.0m on this scale. The disk brightness, S, is proportional to the exposed solar disk area, where E is the solar surface emission in watts/m2 due to the Planck distribution for the solar temperature of T=5770 k.  This results in the formula:

m = -26.5 – 2.5log10(F)

where F is the fraction of the full disk exposed and is equal to Equation 1.  For a sun disk where 90% has been eclipsed, f=0.10 and the dimming is only 2.5log(1/10) = 2.5m. How this translates into how humans perceive ambient lighting is complicated.

The concept of a Just Noticeable Difference is an active research area in psychophysics. In assessing heaviness, for example, the difference between two stimuli of 10 and 11 grams could be detected, but we would not be able to detect the difference between 100 and 101 grams. As the magnitude of the stimuli grow, we need a larger actual difference for detection. The percentage of change remains constant in general. To detect the difference in heaviness, one stimulus would have to be approximately 2 percent heavier than the other; otherwise, we will not be able to spot the difference. Psychologists refer to the percentages that describe the JND as Weber fractions, named after Ernst Weber (1795-1878), a German physiologist whose pioneering research on sensation had a great impact on psychological studies. For example, humans require a 4.8% change in loudness to detect a change; a 7.9% change in brightness is necessary. These values will differ from one person to the next, and from one occasion to the next. However, they do represent generally accurate values.

The minimum perceivable light intensity change is sometimes stated to be 1%, corresponding to +5.0m, but for the Weber Fraction a 7.8% change is required in brightness corresponding to only -2.5log(0.078) = +2.7m. This is compounded by whether the observer is told beforehand that a change is about to happen. If they are not informed, this threshold magnitude dimming could be several magnitudes higher and perhaps closer to the +5.0m value.


The Big Bang: Explained at the reading level of Genesis.

I have often wondered how the modern description of the Big Bang could be written as a story that people at different reading levels would be able to understand, so here are some progressively more complete descriptions beginning with Genesis and their reading level determined by Reliability Formulas.

Genesis (from MIT Bible Gateway)

In the beginning God created the heavens and the earth. Now the earth was formless and empty, darkness was over the surface of the deep, and the Spirit of God was hovering over the waters. And God said, “Let there be light,” and there was light. God saw that the light was good, and he separated the light from the darkness. God called the light “day,” and the darkness he called “night.” And there was evening, and there was morning–the first day. And God said, “Let there be an expanse between the waters to separate water from water.” So God made the expanse and separated the water under the expanse from the water above it. And it was so. God called the expanse “sky.” And there was evening, and there was morning–the second day. And God said, “Let the water under the sky be gathered to one place, and let dry ground appear.” And it was so. God called the dry ground “land,” and the gathered waters he called “seas.” And God saw that it was good. Then God said, “Let the land produce vegetation: seed-bearing plants and trees on the land that bear fruit with seed in it, according to their various kinds.” And it was so. The land produced vegetation: plants bearing seed according to their kinds and trees bearing fruit with seed in it according to their kinds. And God saw that it was good. And there was evening, and there was morning–the third day. And God said, “Let there be lights in the expanse of the sky to separate the day from the night, and let them serve as signs to mark seasons and days and years, and let them be lights in the expanse of the sky to give light on the earth.” And it was so. God made two great lights–the greater light to govern the day and the lesser light to govern the night. He also made the stars. God set them in the expanse of the sky to give light on the earth, to govern the day and the night, and to separate light from darkness. And God saw that it was good.”

The Flesch Reading ease Score gives this an 87.9 ‘easy to read‘ score. Flesch-Kincaid gives this a grade level of 4.5. The Automated Readability Index gives it an index of 4 which is 8-9 year olds in grades 4-5. Amazingly, the scientific content in this story is completely absent and in fact promotes many known misconceptions appropriate to what children under age-5 know about the world.

Can we do at least as well as this story in a 365-word summary that describes the origin of the universe, the origin of the sun, moon and earth, and the appearance of life? Because the reading level of Genesis is only at most Grade-5, can we describe a scientific treatment using only concepts known by the average Fifth-Grader? According to the Next Generation Science Standards, students know about gravity, and scales of time but ideas about atoms and other forces are for Grade 6 and above. The average adult reader can fully comprehend a text with a reading grade level of eight. So if the text has an eighth grade Flesch Kincaid level, its text should be easy to read and accessible by the average US adult. But according to Wylie Communications, half of all US adults read at or below 8th-grade level. The American Academy of Arts and Sciences survey also shows that US adults know about atoms (51%), that the universe began with a Big Bang (41%) and that Earth orbits the sun (76%) so that US adults rank between 5th and 9th internationally in our basic scientific knowledge.

The genesis story splits itself into three distinct parts: The origin of the universe;The origin of stars and planets; and The origin of life and humanity. Only the middle story has detailed observational evidence at every stage. The first and last stories were one-of events for which exact replication and experimentation is impossible.

Because we are 3000 years beyond the writing of Genesis, let’s allow a 400-word limit for each of these three parts and aim at a reading level and science concept level not higher than 7th grade.

First try (497 words):

Origin of the Universe. Our universe emerged from a timeless and spaceless void. We don’t know what this Void is, only that it had none of the properties we can easily imagine. It had no dimension, or space or time; energy or mass; color or absence of color. Scientists use their mathematics to imagine it as a Pure Nothingness. Not even the known laws of nature existed.

Part of this Void exploded in a burst of light and energy that expanded and created both time and space as it evolved in time. This event also locked into existence what we call the Laws of Nature that describe how many dimensions exist in space, the existence of four fundamental forces, and how these forces operate through space and time.

At first this energy was purely in the form of gravity, but as the universe cooled, some of this energy crystalized into particles of matter. Eventually, the familiar elementary particles such as electrons and quarks emerged and this matter became cold enough that basic elements like hydrogen and helium could form.

But the speed at which the universe was expanding wasn’t steady in time. Instead this expansion doubled in speed so quickly that within a fraction of a second, the space in our universe inflated from a size smaller than a baseball to something many billions of miles across. Today, after 14 billion years of further expansion we see only a small fraction of this expanded space today, and we call it the Observable Universe. But compared to all the space that came out of the Big Bang, our entire Observable Universe is as big as a grain of sand compared to the size of our Earth. The Universe is truly an enormous collection of matter, radiation and energy in its many forms.

Meanwhile, the brilliant ‘fireball’ light from the Big Bang also cooled as the universe expanded so that by one million years after the Big Bang, it was cooler than the light we get from the surface of our own sun. Once this light became this cool, familiar atoms could start to form. As the universe continued to expand and cool, eventually the light from the Big Bang became so cool that it could only be seen as a dull glow of infrared light every where in space. The atoms no longer felt the buffeting forces of this fireball light and had started to congregate under the force of gravity into emmence clouds throughout space. It is from these dark clouds that the first stars would begin to form.

Mixed in with the ordinary matter of hydrogen and helium atoms was a mysterious new kind of matter. Scientists call this dark matter because it is invisible but it still affects normal matter by its gravity. Dark matter in the universe is five times more common than ordinary matter. It prevents galaxies like the Milky Way from flying apart, and clusters of galaxies from dissolving into individual galaxies.

Flesch Reading Ease 60. (Average difficulty); Flesch-Kincaid Grade: 10.2; Automated Readability Index: 11

Second Try (538 words):

Origin of the Universe. Our universe emerged from a timeless and spaceless void. We don’t know what this Void was. We think it had none of the properties we can easily imagine. It had no dimension, or space or time. It had no energy or mass. There was no color to it either blackness or pure white. Scientists use their mathematics to imagine it as a Pure Nothingness. They are pretty sure that not even the known laws of nature existed within this Void.

Part of this Void exploded in a burst of light and energy. Astronomers call this the Big Bang. It  expanded and created both time and space as it evolved in time. This event also locked into existence what we call the Laws of Nature. These Laws describe how many dimensions exist in space. The Laws define the four fundamental forces, and how they operate through space and time.

At first the energy in the Big bang was purely in the form of gravity. But as the universe expanded and cooled, some of this energy crystalized into particles of matter. Eventually, the familiar elementary particles such as electrons and quarks emerged. This matter became cold enough that basic elements like hydrogen and helium could form.

But the speed at which the universe was expanding wasn’t steady in time. Instead this expansion doubled in speed very quickly. Within a fraction of a second, the space in our universe grew from a size smaller than a baseball to something many billions of miles across. After 14 billion years of further expansion we see only a small fraction of this expanded space today. We call it the Observable Universe. But compared to all the space that came out of the Big Bang, our entire Observable Universe is as big as a grain of sand compared to the size of our Earth. The Universe is truly an enormous collection of matter, radiation and energy in its many forms.

Meanwhile, the brilliant ‘fireball’ light from the Big Bang also cooled as
the universe expanded. By one million years after the Big Bang, it was cooler than the light we get from the surface of our own sun. Once this light became this cool, familiar atoms could start to form. As the universe continued to expand and cool, eventually the blinding light from the Big Bang faded into a dull glow of infrared light. At this time, a human would see the universe as completely dark. The atoms no longer felt the buffeting forces of this fireball light. They began to congregate under the force of gravity. Within millions of years, immense clouds began to form throughout space. It is from these dark clouds that the first stars would begin to form.

Mixed in with the ordinary matter of hydrogen and helium atoms was a mysterious new kind of matter. Scientists call this dark matter.  It is invisible to the most powerful telescopes, but it still affects normal matter by its gravity. Dark matter in the universe is five times more common than the ordinary matter we see in stars. It prevents galaxies like the Milky Way from flying apart. It also prevents clusters of galaxies from dissolving into individual galaxies.

Flesch Reading Ease 66.7. (Average difficulty); Flesch-Kincaid Grade: 7.2; Automated Readability Index: 6.6 (11-13 year olds).

Third Try ( 410 words )

Origin of the Universe. Our universe appeared out of a timeless and spaceless void. We don’t know what this Void was. We can’t describe it by its size, its mass or its color.  It wasn’t even ‘dark’  because dark (black) is a color.  Scientists think of it as a Pure Nothing.

Part of this Void exploded in a burst of light and energy. We don’t know why.  Astronomers call this event the rather funny name of the ‘Big Bang’. It  was the birth of our universe. But it wasn’t like a fireworks explosion. Fireworks expand into the sky, which is space that already exists. The Big Bang created space as it went along.  There was nothing for it to expand into. The Big Bang also created  what we call the Laws of Nature. These Laws describe how forces like gravity and matter affect each other.

As the universe expanded and cooled, some of its energy became particles of matter. This is like raindrops condensing from a cloud when the cloud gets cool enough. Over time, these basic particles  formed  elements like hydrogen and helium.

The universe continued to expand. Within the blink of an eye, it grew from a size smaller than a baseball to something many billions of miles across. Today, after 14 billion years  we see only a small piece of this expanded space today. Compared to all the space that came out of the Big Bang, what we see around us is as big as a grain of sand compared to the size of our Earth. The Universe is truly enormous!

After about one million years  the fireball light from the Big Bang became very dim. At this time, a human would see the universe as completely dark. There were, as yet, no stars to light up the sky and the darkness of space. Atoms  began to congregate under the force of gravity. Within millions of years, huge clouds the size of  our entire Milky Way galaxy began to form throughout space. From these dark clouds, the first stars started to appear.

Mixed in with  ordinary matter  was a mysterious new kind of matter. Scientists call this dark matter.  It is invisible to the most powerful telescopes. But it still affects normal matter by its gravity, and that’s a very good thing! Without dark matter,  galaxies like our Milky Way and its billions of stars would fly apart, sending their stars into the dark depths of intergalactic space.

Flesch Reading Ease 73.3. (Fairly easy to read); Flesch-Kincaid Grade: 5.9 (Sixth grade) ; Automated Readability Index: 5.1 (8 – 9 year olds) Fourth to Fifth grade.

Summary.

The Third Try is about as simple and readable a story as I can conjure up, and it comes in at a reading level close to Fourth grade. Scientifically, it works with terms like energy, space, expansion, matter  and gravity, and scales like millions and billions of years. All in all, it is not a bad attempt that reads pretty well, scientifically, and does not mangle some basic ideas. It also has a few ‘gee whiz’ ideas like Nothing, space expansion and dark matter. 

So, what do you think? Leave me a note at my Facebook page!

Next time I will tackle the middle essay about the formation of  stars and planets!

This is Not Your Father’s Universe!

When I was learning astronomy in the 60s and 70s, we were still debating whether Big Bang or Stady State were the most accurate models for our universe. We also wondered about how galaxies like our Milky Way formed, and whether black holes existed. The idea that planets beyond our solar system existed was pure science fiction and no astronomers spent any time trying to predict what they might look like. As someone who has reached the ripe old age of 70, I am amazed how much progress we have made, from the discovery of supermassive black holes and exoplanets, to dark matter and gravitational radiation. The pace of discovery continues to increase, and our theoretical ideas are now getting confirmed or thrown out at record pace. There are still some issues that remain deliciously mysterious. Here are my favorite Seven Mysteries of the Universe!

1-How to Build a Galaxy: In astronomy, we used to think that it would take the universe a long time to build galaxies like our Milky Way, Thanks to the new discoveries by the Webb Space Telescope, we now have a ring-side seat to how this happens, and boy is it a fast process!

The oldest galaxies discovered with the Hubble Space Telescope date back to between 400 and 500 million years after the Big Bang. A few weeks ago, Webb spotted a galaxy that seems to have formed only 300 million years after the Big Bang. Rather than the massive galaxies like the Milky Way, these young galaxies resemble the dwarf galaxies like the Large Magellanic Cloud, perhaps only 1/10 the mass of our galaxy and filled with enormus numbers of massive, luminous stars. The above image from Hubble is a nearby galaxy called M-33 that has a mass of about 50 billion suns. There were lots of these smaller galaxies being formed during the first 300 million years after the Big Bang.

It looks like the universe emerged from the Dark Ages and immediately started building galaxies. In time, these fragments collided and merged to become the more massive galaxies we see around us, so we are only just starting to see how galaxy-building happens. Our Milky Way was formed some 1 billion years after the Big Bang, so the galaxy fragments being spotted by Webb have another 700 million years to go to make bigger things. Back in 2012, Hubble had already discovered the earliest spiral galaxy seen by then; a galaxy called Q2343-BX442, camping out at 3 billion years after the Big Bang. In 2021, an even younger spiral galaxy was spotted, called BRI 1335-0417, seen as it was about 1.4 billon years after the Big Bang.

So we are now watching how galaxies are being formed almost right before our eyes! Previous ideas that I learned about as an undergraduate, in which galaxies are formed ‘top-down’ from large collections of matter that fragment into stars, now seem wrong or incomplete. The better idea is that smaller collections of matter form stars and then merge together to build larger systems – called the ‘bottom-up’ model. This process is very, very fast! Among the smallest of these ‘galaxies’ are things destined to become the globular clusters we see today.

2-Supermassive Black Holes: The most distant and youngest supermassive black hole was discovered in 2021. Called J0313-1806, its light left it to reach us when the universe was only 670 million years old. Its mass, however, is a gargantuan 1.6 BILLION times the mass of our sun. Even if the formation of this black hole started at the end of the Dark Ages ca 100 million years after the Big Bang, it would have to absorb matter at the rate of three suns every year on average. That explains its quasar energy, but still…it is unimaginable how these things can grow so fast! The only working idea is that they started from seed masses about 10,000 the mass of our sun and grew from there. But how were the seed masses formed? This remains a mystery today.

3-The Theory of everything: The next Big Thing that I have been following since the 1960s is the search for what some call the Theory of Everything. Exciting theoretical advancements were made in the 1940s and 1960s to create accurate mathematical models for the three nongravity forces, called the electromagnetic, weak and strong forces. Physicists call this the Standard Model, and every physicist learns its details as students in graduate school. By the early 1980s, string theory was able to add gravity to the mix and go beyond the Standard Model. It appeared that the pursuit of a unified theory had reached its apex. Fifty years later, this expectation has all but collapsed.

Experiments at the Large Hadron Collider continue to show how the universe does not like something called supersymmetry in our low-energy universe. Supersymmetry is a key ingredient to string theory because it lets you change one kind of particle (field) into another, which is a key ingredient to any unified theory. So the simplest versions of string theory rise or fall based on whether supersymmetry exists or not. For over a decade, physicists have tried to find places in the so-called Standard Model where supersymmetry should make its appearance, but it has been a complete no-show. Its not just that this failure is a problem for creating a more elegant theory of how forces works, but it also affects astronomy as well.

https://penntoday.upenn.edu/news/making-sense-string-theory

Personally, when string theory hit the stage in the 1980’s I, like many other astronomers and physicists, thought that we were on the verge of solving this challenge of unifying gravity with the other forces, but this has not been the reality. Even today, I see no promissing solutions to this vexing problem since apparently the data shows that simple string theory is apparently on a wrong theoretical track.

One cheerful note: For neutrinos, the path from theoretical prediction to experimental observation took 25 years. For the Higgs boson, it took 50 years. And for gravitational waves, it took a full 100 years. We may just have to be patient…for another 100 years!

4-Dark Matter: The biggest missing ingredient to the cosmos today is called dark matter. When astronomers ‘weigh’ the universe, they discover that 4.6% of its gravitating ‘stuff’ is in ordinary matter (atoms,. stars, gas, neutron stars etc), but a whopping 24% is in some other ‘stuff’ that only appears by its gravity. It is otherwise completely invisible. Putting this another way, it’s as though four out of every five stars that make up our Milky Way were completely invisible.

Dark matter in Abell 1679.

Because we deeply believe that dark matter must be tracable to a new kind of particle, and because the Standard Model gives us an accounting of all the kinds of elementary particles from which our universe is built, the dark matter particle has to be a part of the Standard Model…but it isn’t!!!! Only by extending the Standard Model to a bigger theory (like string theory) can we logically and mathematically add new kinds of particles to a New Standard Model- one of which would be the dark matter particle. String theory even gives us a perfect candidate called the neutralino! But the LHC experiments have told us for over a decade that there is nothing wrong with the Standard Model and no missing particles. Astronomers say that dark matter is real, but physicists can’t find it….anywhere. Well…maybe not ALL astronomers think it’s real. So the debate continues.

Personally, I had heard about ‘missing mass’ in the 1960s but we were all convinced we would find it in hot gas, dim red dwarf stars or even black holes. I NEVER thought that it would turn out to be something other than ordinary matter in an unusual form. Dark matter is so deeply confounding to me that I worry we will not discover its nature before I, myself, leave this world! Then again, there isnt a single generation of scientists that has had all its known puzzles neatly solved ‘just in time’. I’m just greedy!!!

5- Matter and Antimatter: During the Big Bang, there were equal amounts of matter and anti matter, but then for some reason all the antimatter dissappeared leaving us with only matter to form atoms, stars and galaxies. We don’t know why this happened, and the Standard Model is completely unhelpful in giving us any clues to explain this. But next to dark matter, this is one of the most outstanding mysteries of modern, 21st century cosmology. We have no clue how to account for this fact within the Standard Model, so again like Dark Matter, we see that at cosmological scales, the Standard Model is incomplete.

6- Origin of Time and Space: Understanding the nature of time and space, and trying to make peace with why they exist at all, is the bane of any physicists existence. I have written many blogs on this subject, and have tried to tackle it from many different angles, but in the end they are like jigsaw puzzels with too many missing pieces. Still, it is very exciting to explore where modern physics has taken us, and the many questions such thinking has opened up in surprising corners. My previous blog about ‘What is ‘Now’ is one such line of thinking. Many of the new ideas were not even imagined as little as 30 years ago, so that is a positive thing. We are still learning more about these two subjects and getting better at asking the right questions!

https://iopscience.iop.org/journal/0264-9381/page/Focus-issue-loop-quantum-gravity

7-Consciousness: OK…You know I would get to this eventually, and here it is! Neuroscientists know of lots of medical conditions that can rob us of consciousness including medical anesthesia, but why we have this sense about ourselves that we are a ‘person’ and have volition is a massively hard problem. In fact, consciousness is called the ‘Hard Problem’ in neuroscience..heck…in any science!

https://www.technologynetworks.com/neuroscience/articles/what-if-consciousness-is-not-what-drives-the-human-mind-307159

The ‘Soft Problem’ is how our senses give us a coherant internal model of the world that we can use to navigate the outside world. We know how to solve the Soft Problem, just follow the neurons. We are well on our way to understanding it thanks to high-tech brain imaging scanners and cleverly-designed experiments. The Hard Problem is ‘hard’ because our point-of-view is within the thing we are trying to undertand. Some think that our own ‘wet ware’ is not up to the task of even giving us the intelligence to answer this qustion. It will not be the first time someone has told us about limitations, but usually these are technological ones, and not ones related to limits to what our own brains can provide as a tool.

So there you have it.

My impression is that only Mysteries #1 and #2 will make huge progress. The Theory of Everything is in experimental disarray. For antimatter, there has been no progress, but many ideas. They all involve going outside the Standard Model. Dark matter might be replaced by a modification of gravity at galactic and cosmological scales.

Beyond these ‘superficial’ mysteries, we are left with three deep mysteries. The origin of space, time and consciousness remain our 21st century gift to children of the 22nd century!

Check back here in a few weeks for the next blog!

What is ‘Now’?

What is the duration of the present moment? How is it that this present moment is replaced by ‘the next moment’?

Within every organism, sentient or not, there are thousands of chemical processes that occur with their own characteristic time periods, but these time periods start and stop at different times so that there is no synchronized ‘moment’. Elementary atomic collisions that build up molecules take nanoseconds while cell division takes minutes to hours, and tissue cell lifespans vary from 2 days in the stomach lining to 8 years for fat cells (see Cell Biology). None of these jangled timescales collectively or in isolation create the uniform experience we have of now and its future moments. To find the timescale that corresponds to the Now experience we have to look elsewhere.

It’s all in the mind!

A variety of articles over the  years have identified 2 to 3 seconds as the maximum duration of what most people experience as ‘now’, and what researchers call the ‘specious present’. This is the time required by our brain’s neurological mechanisms to combine the information arriving at our senses with our internal, current model of the ‘outside world’. During this time an enormous amount of neural activity has to happen. Not only does the sensory information have to be integrated together for every object in your visual field and cross connected to the other senses, but dozens of specialized brain regions have to be activated or de-activated to update your world model in a consistent way.

In a previous blog I discussed how important this world model is in creating within you a sense of living in a consistent world with a coherent story. But this process is not fixed in stone. Recent studies by Sebastian Sauer and his colleagues at the Ludwig-Maximilians-Universität in Munich show that mindfulness meditators can significantly increase their sense of ‘now’ so that it is prolonged for up to 20 seconds.

In detail, a neuron discharge lasts about 1 millisecond, but it has to be separated from the next one by about 30 milliseconds before a sequence is perceived, and this seems to be true for all senses. When you see a ‘movie’ it is a succession of still images flashed into your visual cortex at intervals less than 30 milliseconds, giving the illusion of a continuous unbroken scene.  (Dainton: Stanford Encyclopedia of Philosophy, 2017).

The knitting together of these ‘nows’ into a smooth flow-of-time is done by our internal model-building system. It works lightning-fast to connect one static collection of sensory inputs to another set and hold these both in our conscious ‘view’ of the world. This gives us a feeling of the passing of one set of conditions smoothly into another set of conditions that now make up the next ‘Now’. To get from one moment to the next, our brain can play fast-and-loose with the data and interpolate what it needs. For example, it our visual world, the fovea in our retina produces a Blind Spot but you never notice it because there are circuits that interpolate across this spot to fill-in the scenery. The same thing happens in the time dimension with the help of our internal model to make our jagged perceptions in time into a smooth movie experience.

Neurological conditions such as strokes, or psychotropic chemicals can disrupt this process and cause dramatic problems. Many schizophrenic patients stop perceiving time as a flow of  linked events.  These defects in time perception may play a part in the hallucinations and delusions experienced by schizophrenic patients according to some studies. There are other milder aberrations that can affect our sense of the flow-of-time.

Research has also suggested the feeling of awe has the ability to expand one’s perceptions of time availability. Fear also produces time-sense distortion. Time seems to slow down when a person skydives or bungee jumps, or when a person suddenly and unexpectedly senses the presence of a potential predator or mate. Research also indicates that the internal clock, used to time durations in the seconds-to-minutes range, is linked to dopamine function in the basal ganglia. Studies in which children with ADHD are given time estimation tasks shows that time passes very slowly for them.

Because the volume of data is enormous, we cannot hold many of these consecutive Now moments in our consciousness with the same clarity, and so earlier Nows either pass into short-term memory if they have been tagged with some emotional or survival attributes, or fade quickly into complete forgetfulness. You will not remember the complete sensory experience of diving into a swimming pool, but if you were pushed, or were injured, you will remember that specific sequence of moments with remarkable clarity years later!

The model-building aspect of our brain is just another tool it has that is equivalent to its pattern-recognition ability in space. It looks for patterns in time to find correlations which it then uses to build up expectations for ‘what comes next’. Amazingly, when this feature yields more certainty than the evidence of our senses, psychologists like Albert Powers at Yale University say that we experience hallucinations (Fan, 2017). In fact, 5-15% of the population experience auditory hallucinations (songs, voices, sounds) at some time in their lives when the brain literally hears a sound that is not there because it was strongly expected on the basis of other clues. One frequent example is that  people claim to hear the Northern Lights as a crackling fire or a swishing sound, because their visual system creates this expectation and the brain obliges.

This, then, presents us with the neurological experience of Now. It is between 30 milliseconds and several minutes in duration. It includes a recollection of the past which fades away for longer intervals in the past, and includes a sense of the immediate future as our model-making facility extrapolates from our immediate past and fabricates an expectation of what comes next.

Living in a perpetual Now is no fun. The famed psychologist Oliver Sacks describes  a patient, Clive Wearing, with a severe form of amnesia, who was unable to form any new memories that lasted longer than 30 seconds, and became convinced every few minutes that he was fully conscious for the first time. “In some ways, he is not anywhere at all; he has dropped out of space and time altogether. He no longer has any inner narrative; he is not leading a life in the sense that the rest of us do….It is not the remembrance of things past, the “once” that Clive yearns for, or can ever achieve. It is the claiming, the filling, of the present, the now, and this is only possible when he is totally immersed in the successive moments of an act. It is the “now” that bridges the abyss.”

Physical ‘Now’.

This monkeying around with brain states, internal model-making and sensory data creates Now as a phenomenon we experience, but the physical world outside our collective brain population does not operate through its own neural systems to create a Cosmic Now. That would only be the case if, for example, we were literally living inside The Matrix….which I believe we are not. So in terms of physics, the idea of Now does not exist. We even know from relativity that there can be no uniform and simultaneous Now spanning large portions of space or the cosmos. This is a problem that has bedeviled many people across the millennia.

Augustine (in the fourth century) wrote, “What is time? If no one asks me, I know; if I wish to explain, I do not know. … My soul yearns to know this most entangled enigma.” Even Einstein himself noted ‘…that there is something essential about the Now which is just outside the realm of science.’

Both of these statements were made before quantum theory became fully developed. Einstein developed relativity, but this was a theory in which spacetime took the place of space and time individually. If you wanted to define ‘now’ by a set of simultaneous conditions, relativity put the kibosh on that idea because due to the relative motions and accelerations of all Observers, there can be no simultaneous ‘now’ that all Observers can experience. Also, there was no ‘flow of time’ because relativity was a theory of worldlines and complete histories of particles from start to finish (called the boundary conditions of worldlines). Quantum theory, however, showed some new possibilities.

In physics, time is a variable, often represented by the letter t, that is a convenient parameter with which to describe how a system of matter and energy change. The first very puzzling feature of time as a physical variable is that all mathematical representations of physical laws or theories show that time is continuous, smooth and infinitely divisible into smaller intervals. These equations are also ‘timeless’ in that they can be written down on a piece of paper and accurately describe how a system changes from start to finish (based on boundary conditions defined at ‘t=0’) , but the equations show this process as ‘all at once’.

In fact, this perspective is so built into physics that it forms the core of Einstein’s relativity theory in the form of the 4-d spacetime ‘block’. It also appears in quantum mechanics because fundamental equations like Schroedinger’s Equation also offer a timeless view of quantum states.

In all these situations, one endearing feature of our world is actually suppressed and mathematically hidden from view, and that is precisely the feature we call ‘now’.

To describe what things look like Now, you have to dial in to the equations the number t =  t(now). How does nature do that? As discussed by physicist Lee Smolin in his book ‘Time Reborn’, this is the most fundamental experience we have about the physical world as sentient beings, yet it is not represented by any feature in the physical theories we have developed thus far. There is no theory that selects t = t(now) over all the infinite other moments in time.

Perhaps we are looking in the wrong place!

Just as we have seen that what we call ‘space’ is built up like a tapestry from a vast number of quantum events described (we hope!) by quantum gravity, time also seems to be created from a synthesis of elementary events occurring at the quantum scale.   For example, what we call temperature is the result of innumerable collisions among elementary objects such as atoms. Temperature is a measure of the average collision energy of a large collection of particles, but cannot be identified as such at the scale of individual particles. Temperature is a phenomenon that has emerged from the collective properties of thousands or trillions of individual particles.

A system can be described completely by its quantum state – which is a much easier thing to do when you have a dozen atoms than when you have trillions, but the principle is the same. This quantum state describes how the elements of the system are arrayed in 3-d space, but because of Heisenberg’s Uncertainty Principle, the location of a particle at a given speed is spread out rather than localized to a definite position.  But quantum states can also become entangled. For these systems, if you measure one of the particles and detect property P1 then the second particle must have property P2. The crazy thing is that until you measured that property in the first particle, it could have had either property P1 or P2, but after the measurement the distant particle ‘knew’ that it had to have the corresponding property even though this information had to travel faster than light to insure consistency.

An intriguing set of papers by physicist Seth Lloyd at Harvard University in 1984 showed that over time, the quantum states of the member particles become correlated and shared by the larger ensemble. This direction of increasing correlation goes only one way and establishes the ‘Arrow of Time’ on the quantum scale.

One interesting feature of this entanglement idea is that ‘a few minutes ago’, our brain’s quantum state was less correlated with its surroundings and our sensory information than at a later time. This means that the further you go into the past moments, the less correlated they are with the current moment because, for one, the sensory information has to arrive and be processed before it can change our brain’s state. Our sense of Now is the product of how past brain states are correlated with the current state. A big part of this correlating is accomplished, not by sterile quantum entanglement, but by information transmitted through our neural networks and most importantly our internal model of our world – which is a dynamic thing.

If we did not have such an internal model that correlates our sensory information and fabricates an internal story of perception, our sense of Now would be very different because so much of the business of correlating quantum information would not occur very quickly. Instead of a Now measured in seconds, our Now’s would be measured in hours, days or even lifetimes, and be a far more chaotic experience because it would lack a coherent, internal description of our experiences.

This seems to suggest that no two people live in exactly the same Now, but these separate Now experiences can become correlated together as the population of individuals interact with each other and share experiences through the process of correlation. As for the rest of the universe, it exists in an undefined Now state that varies from location to location and is controlled by the speed of light, which is the fastest mode of exchanging information.

Read more:

In my previous blogs, I briefly described how the human brain perceives and models space (Blog 14: Oops one more thing), how Einstein and other physicists dismiss space as an illusion (Blog 10: Relativity and space ), how relativity deals with the concept of space (Blog 12: So what IS space?), what a theory of quantum gravity would have to look like (Blog 13: Quantum Gravity Oh my! ), and along the way why the idea of infinity is not physically real (Blog 11: Is infinity real?) and why space is not nothing (Blog 33: Thinking about Nothing). I even discussed how it is important to ‘think visually’ when trying to model the universe such as the ‘strings’ and ‘loops’ used by physicists as an analog to space ( Blog 34: Thinking Visually)

I also summarized the nature of space in a wrap-up of why something like a quantum theory for gravity is badly needed because the current theories of quantum mechanics and general relativity are incomplete, but also point the way towards a theory that is truly background-independent and relativistic (Blog 36: Quantum Gravity-Again! ). These considerations describe the emergence of the phenomenon we call ‘space’ but also down play its importance because it is an irrelevant and misleading concept.

Black Holes for Fun and Profit!

Well, astronomers finally did it. Nearly 100 years ago, Albert Einstein’s theory of General Relativity predicted that black holes should exist. Although it took until the 1960’s for someone like physicist John Wheeler to coin the name ‘black hole’ the study of these enigmatic objects became a cottage industry in theoretical physics and astrophysics. In fact, for certain kinds of astronomical phenomena such as quasars and x-ray sources, there was simply no other explanation for how such phenomena could generate so much energy in such an impossibly small volume of space. The existence of black holes was elevated to a certainty during the 1990s as studies of distant galaxies by the Hubble Space Telescope turned in tons of data that clinched the idea that the cores of most if not all galaxies had them. In fact, these black holes contained millions or even billions of times the mass of our sun and were awarded a moniker all their own: supermassive black holes. But there was still one outstanding problem for these versatile engines of gravitational destruction: Not a single one had ever been seen. To understand why, we have to delve rather deeply into what these beasties really are. Hang on to your seats!

               General relativity is the preeminent theory of gravity, but it is completely couched in the language of geometry – in this case the geometry of what is called our 4-dimensional spacetime continuum. You see, in general relativity, what we call space is just a particular feature of the gravitational field of the cosmos within which we are embedded rather seamlessly. This is all well and good, and this perspective has led to the amazing development of the cosmological model called Big Bang theory. Despite this amazing success, it is a theory not without its problems. The problems stem from what happens when you collect enough mass together in a small volume of space so that the geometry of spacetime (e.g. the strength of gravity) becomes enormously curved.

               The very first thing that happens according to the theory is that a condition in spacetime called a Singularity forms. Here, general relativity itself falls apart because density and gravity tend towards infinite conditions. Amazingly according to general relativity, and proved by the late Stephen Hawkins, spacetime immediately develops a zone surrounding the Singularity called an event horizon. For black holes more massive than our sun, the distance in kilometers of this spherical horizon from the Singularity is just 2.9 times the mass of the black hole in multiples of the sun’s mass. For example, if the mass of a supermassive black hole is 6.5 billion times the mass of our sun, its event horizon is at 6.5 billionx2.9 or 19 billion kilometers. Our solar system has a radius of only 8 billion km to Pluto, so this supermassive black hole is over twice the size of our solar system!

               Now the problem with event horizons is that they are one-way. Objects and even light can travel through them from outside the black hole, but once inside they can never return to the outside universe to give a description of what happened. However, it is a misunderstanding to say that black holes ‘suck’ as the modern colloquialism goes. They are simply points of intense gravitational force, and if our sun were replaced by one very gently, our Earth would not even register the event and continue its merry way in its orbit. The astrophysicist’s frustration is that it has never been possible to take a look at what is going on around the event horizon…until April 10, 2019.

               Researchers using the radio telescope interferometer system called the Event Horizon Telescope were able to synthesize an image of the surroundings of the supermassive black hole in the quasar-like galaxy Messier-87 – also known as Virgo A by early radio astronomers after World War II, and located about 55 million light years from Earth. They combined the data from eight radio telescopes scattered from Antarctica to the UK to create one telescope with the effective diameter of the entire Earth. With this, they were able to detect and resolve details at the center of M-87 near the location of a presumed supermassive black hole. This black hole is surrounded by a swirling disk of magnetized matter, which ejects a powerful beam of plasma into intergalactic space. It has been intensively studied for decades and the details of this process always point to a supermassive black hole as the cause.

Beginning in 2016, several petabytes of data were gathered from the Event Horizon Telescope and a massive press conference was convened to announce the first images of the vicinity of the event horizon. Surrounding the black shadow zone containing the event horizon was a clockwise-rotating ring of billion-degree plasma traveling at nearly the speed of light. When the details of this image were compared with supercomputer simulations, the mass of the supermassive black hole could be accurately determined as well as the dynamics of the ring plasma. The round shape of the event horizon was not perfect, which means that it is a rotating Kerr-type black hole. The darkness of the zone indicated that the event horizon did not have a photosphere of hot matter like the surface of our sun, so many competing ideas about this mass could be eliminated. Only the blackness of a black hole and its compact size now remain as the most consistent explanation for what we are ‘seeing’. Over time, astronomers will watch as this ring plasma moves from week to week. The next target for the Event Horizon Telescope is the four million solar mass black hole at the center of the Milky Way called Sgr-A*. Watch this space for more details to come!!

We have truly entered a new world in exploring our universe. Now if someone could only do something about dark matter!!!

Decay of the False Vacuum

The Decay of the False Vacuum

Written by Sten Odenwald. Copyright (C) 1983 Kalmbach Publishing. Reprinted by permission

In the recently developed theory by Steven Weinberg and Abdus Salam, that unifies the electromagnetic and weak forces, the vacuum is not empty. This peculiar situation comes about because of the existence of a new type of field, called the Higgs field. The Higgs field has an important physical consequence since its interaction with the W, W and Z particles (the carriers of the weak force) causes them to gain mass at energies below 100 billion electron volts (100 Gev). Above this energy they are quite massless just like the photon and it is this characteristic that makes the weak and electromagnetic forces so similar at high energy.

On a somewhat more abstract level, consider Figures 1 and 2 representing the average energy of the vacuum state. If the universe were based on the vacuum state in Figure 1, it is predicted that the symmetry between the electromagnetic and weak interactions would be quite obvious. The particles mediating the forces would all be massless and behave in the same way. The corresponding forces would be indistinguishable. This would be the situation if the universe had an average temperature of 1 trillion degrees so that the existing particles collided at energies of 100 Gev. In Figure 2, representing the vacuum state energy for collision energies below 100 Gev, the vacuum state now contains the Higgs field and the symmetry between the forces is suddenly lost or ‘broken’. Although at low energy the way in which the forces behave is asymmetric, the fundamental laws governing the electromagnetic and weak interactions remain inherently symmetric. This is a very remarkable and profound prediction since it implies that certain symmetries in Nature can be hidden from us but are there nonetheless.

During the last 10 years physicists have developed even more powerful theories that attempt to unify not only the electromagnetic and weak forces but the strong nuclear force as well. These are called the Grand Unification Theories (GUTs) and the simplist one known was developed by Howard Georgi, Helen Quinn,and Steven Weinberg and is called SU(5), (pronounced ‘ess you five’). This theory predicts that the nuclear and ‘electroweak’ forces will eventually have the same strength but only when particles collide at energies above 1 thousand trillion GeV corresponding to the unimaginable temperature of 10 thousand trillion trillion degrees! SU(5) requires exactly 24 particles to mediate forces of which the 8 massless gluons of the nuclear force, the 3 massless intermediate vector bosons of the weak force and the single massless photon of the electromagnetic force are 12. The remaining 12 represent a totally new class of particles called Leptoquark bosons that have the remarkable property that they can transform quarks into electrons. SU(5) therefore predicts the existence of a ‘hyperweak’ interaction; a new fifth force in the universe! Currently, this force is 10 thousand trillion trillion times weaker than the weak force but is nevertheless 100 million times stronger than gravity. What would this new force do? Since protons are constructed from 3 quarks and since quarks can now decay into electrons, through the Hyperweak interaction, SU(5) predicts that protons are no longer the stable particles we have always imagined them to be. Crude calculations suggest that they may have half-lives between 10^29 to 10^33 years. An immediate consequence of this is that even if the universe were destined to expand for all eternity, after ‘only’ 10^32 years or so, all of the matter present would catastrophically decay into electrons, neutrinos and photons. The Era of Matter, with its living organisms, stars and galaxies, would be swept away forever, having represented but a fleeting episode in the history of the universe. In addition to proton decay, SU(5) predicts that at the energy characteristic of the GUT transition, we will see the affects of a new family of particles called supermassive Higgs bosons whose masses are expected to be approximately 1 thousand trillion GeV! These particles interact with the 12 Leptoquarks and make them massive just as the Higgs bosons at 100 GeV made the W, W and Z particles heavy. Armed with this knowledge, let’s explore some of the remarkable cosmological consequences of these exciting theories.

The GUT Era

To see how these theories relate to the history of the universe, imagine if you can a time when the average temperature of the universe was not the frigid 3 K that it is today but an incredable 10 thousand trillion trillion degrees (10^15 GeV). The ‘Standard Model’ of the Big Bang, tells us this happened about 10^-37 seconds after Creation. The protons and neutrons that we are familiar with today hadn’t yet formed since their constituent quarks interacted much too weakly to permit them to bind together into ‘packages’ like neutrons and protons. The remaining constituents of matter, electrons, muons and tau leptons, were also massless and traveled about at essentially light-speed; They were literally a new form of radiation, much like light is today! The 12 supermassive Leptoquarks as well as the supermassivs Higgs bosons existed side-by-side with their anti-particles. Every particle-anti particle pair that was annihilated was balanced by the resurrection of a new pair somewhere else in the universe. During this period, the particles that mediated the strong, weak and electromagnetic forces were completely massless so that these forces were no longer distinguishable. An inhabitant of that age would not have had to theorize about the existence of a symmetry between the strong, weak and electromagnetic interactions, this symmetry would have been directly observable and furthermore, fewer types of particles would exist for the inhabitants to keep track of. The universe would actually have beed much simpler then!

As the universe continued to expand, the temperature continued to plummet. It has been suggested by Demetres Nanopoulis and Steven Weinberg in 1979 that one of the supermassive Higgs particles may have decayed in such a way that slightly more matter was produced than anti-matter. The remaining evenly matched pairs of particles and anti-particles then annihilated to produce the radiation that we now see as the ‘cosmic fireball’.

Exactly what happened to the universe as it underwent the transitions at 10^15 and 100 GeV when the forces of Nature suddenly became distinguishable is still under investigation, but certain tantalizing descriptions have recently been offered by various groups of theoriticians working on this problem. According to studies by Alan Guth, Steven Weinberg and Frank Wilczyk between 1979 and 1981, when the GUT transition occured, it occured in a way not unlike the formation of vapor bubbles in a pot of boiling water. In this analogy, the interior of the bubbles represent the vacuum state in the new phase, where the forces are distinguishable, embedded in the old symmetric phase where the nuclear, weak and electromagnetic forces are indistinguishable. Inside these bubbles, the vacuum energy is of the type illustrated by Figure 2 while outside it is represented by Figure 1. Since we are living within the new phase with its four distinguishable forces, this has been called the ‘true’ vacuum state. In the false vacuum state, the forces remain indistinguishable which is certainly not the situation that we find ourselves in today!

Cosmic Inflation

An exciting prediction of Guth’s model is that the universe may have gone through at least one period in its history when the expansion was far more rapid than predicted by the ‘standard’ Big Bang model. The reason for this is that the vacuum itself also contributes to the energy content of the universe just as matter and radiation do however, the contribution is in the opposite sense. Although gravity is an attractive force, the vacuum of space produces a force that is repulsive. As Figures 1 and 2 show, the minimum energy state of the false vacuum at ‘A’ before the GUT transition is at a higher energy than in the true vacuum state in ‘B’ after the transition. This energy difference is what contributes to the vacuum energy. During the GUT transition period, the positive pressure due to the vacuum energy would have been enormously greater than the restraining pressure produced by the gravitational influence of matter and radiation. The universe would have inflated at a tremendous rate, the inflation driven by the pressure of the vacuum! In this picture of the universe, Einstein’s cosmological constant takes on a whole new meaning since it now represents a definite physical concept ; It is simply a measure of the energy difference between the true and false vacuum states (‘B’ and ‘A’ in Figures 1 and 2.) at a particular time in the history of the universe. It also tells us that, just as in de Sitter’s model, a universe where the vacuum contributes in this way must expand exponentially in time and not linearly as predicted by the Big Bang model. Guth’s scenario for the expansion of the universe is generally called the ‘inflationary universe’ due to the rapidity of the expansion and represents a phase that will end only after the true vacuum has supplanted the false vacuum of the old, symmetric phase.

A major problem with Guth’s original model was that the inflationary phase would have lasted for a very long time because the false vacuum state is such a stable one. The universe becomes trapped in the cul-de-sac of the false vacuum state and the exponential expansion never ceases. This would be somewhat analogous to water refusing to freeze even though its temperature has dropped well below 0 Centigrade. Recent modifications to the original ‘inflationary universe’ model have resulted in what is now called the ‘new’ inflationary universe model. In this model, the universe does manage to escape from the false vacuum state and evolves in a short time to the familiar true vacuum state.

We don’t really know how exactly long the inflationary phase may have lasted but the time required for the universe to double its size may have been only 10^-34 seconds. Conceivably, this inflationary period could have continued for as ‘long’ as 10^-24 seconds during which time the universe would have undergone 10 billion doublings of its size! This is a number that is truely beyond comprehension. As a comparison, only 120 doublings are required to inflate a hydrogen atom to the size of the entire visible universe! According to the inflationary model, the bubbles of the true vacuum phase expanded at the speed of light. Many of these had to collide when the universe was very young in order that the visible universe appear so uniform today. A single bubble would not have grown large enough to encompass our entire visible universe at this time; A radius of some 15-20 billion light years. On the other hand, the new inflationary model states that even the bubbles expanded in size exponentially just as their separations did. The bubbles themselves grew to enormous sizes much greater than the size of our observable universe. According to Albrecht and Steinhardt of the University of Pennsylvania, each bubble may now be 10^3000 cm in size. We should not be too concerned about these bubbles expanding at many times the speed of light since their boundaries do not represent a physical entity. There are no electrons or quarks riding some expandind shock wave. Instead, it is the non-material vacuum of space that is expanding. The expansion velocity of the bubbles is not limited by any physical speed limit like the velocity of light.

GUMs in GUTs

A potential problem for cosmologies that have phase transitions during the GUT Era is that a curious zoo of objects could be spawned if frequent bubble mergers occured as required by Guth’s inflationary model. First of all, each bubble of the true vacuum phase contains its own Higgs field having a unique orientation in space. It seems likely that no two bubbles will have their Higgs fields oriented in quite the same way so that when bubbles merge, knots will form. According to Gerhard t’Hooft and Alexander Polyakov, these knots in the Higgs field are the magnetic monopoles originally proposed 40 years ago by Paul Dirac and there ought to be about as many of these as there were bubble mergers during the transition period. Upper limits to their abundance can be set by requiring that they do not contribute to ‘closing’ the universe which means that for particles of their predicted mass (about 10^16 GeV), they must be 1 trillion trillion times less abundant than the photons in the 3 K cosmic background. Calculations based on the old inflationary model suggest that the these GUMs (Grand Unification Monopoles) may easily have been as much as 100 trillion times more abundant than the upper limit! Such a universe would definitly be ‘closed’ and moreover would have run through its entire history between expansion and recollapse within a few thousand years. The new inflationary universe model solves this ‘GUM’ overproduction problem since we are living within only one of these bubbles, now almost infinitly larger than our visible universe. Since bubble collisions are no longer required to homogenize the matter and radiation in the universe, very few, if any, monopoles would exist within our visible universe.

Horizons

A prolonged period of inflation would have had an important influence on the cosmic fireball radiation. One long-standing problem in modern cosmology has been that all directions in the sky have the same temperature to an astonishing 1 part in 10,000. When we consider that regions separated by only a few degrees in the sky have only recently been in communication with one another, it is hard to understand how regions farther apart than this could be so similar in temperature. The radiation from one of these regions, traveling at the velocity of light, has not yet made it across the intervening distance to the other, even though the radiation may have started on its way since the universe first came into existence. This ‘communication gap’ would prevent these regions from ironing-out their temperature differences.

With the standard, Big Bang model, as we look back to earlier epochs from the present time, the separations between particles decrease more slowly than their horizons are shrinking. Neighboring regions of space at the present time, become disconnected so temperature differences are free to develope. Eventually, as we look back to very ancient times, the horizons are so small that every particle existing then literally fills the entire volume of its own, observable universe. Imagine a universe where you occupy all of the available space! Prior to the development of the inflationary models, cosmologists were forced to imagine an incredably well-ordered initial state where each of these disconnected domains (some 10^86 in number) had nearly identical properties such as temperature. Any departure from this situation at that time would have grown to sizable temperature differences in widely separated parts of the sky at the present time. Unfortunately, some agency would have to set-up these finely-tuned initial conditions by violating causality. The contradiction is that no force may operate by transmitting its influence faster than the speed of light. In the inflationary models, this contradiction is eliminated because the separation between widely scattered points in space becomes almost infinitly small compared to the size of the horizons as we look back to the epoc of inflation. Since these points are now within each others light horizons, any temperature difference would have been eliminated immediatly since hotter regions would now be in radiative contact with colder ones. With this exponentially-growing, de Sitter phase in the universe’s early history we now have a means for resolving the horizon problem.

Instant Flat Space

Because of the exponential growth of the universe during the GUT Era, its size may well be essentially infinite for all ‘practical’ purposes . Estimates by Albrecht and Steinhardt suggest that each bubble region may have grown to a size of 10^3000 cm by the end of the inflationary period. Consequently, the new inflationary model predicts that the content of the universe must be almost exactly the ‘critical mass’ since the sizes of each of these bubble regions are almost infinite in extent. The universe is, for all conceivable observations, exactly Euclidean (infinite and flat in geometry) and destined to expand for all eternity to come. Since we have only detected at most 10 percent of the critical mass in the form of luminous matter, this suggests that 10 times as much matter exists in our universe than is currently detectable. Of course, if the universe is essentially infinite this raises the ghastly spectre of the eventual annihilation of all organic and inorganic matter some 10^32 years from now because of proton decay.

In spite of its many apparent successes, even the new inflationary universe model is not without its problems. Although it does seem to provide explainations for several cosmological enigmas, it does not provide a convincing way to create galaxies. Those fluctuations in the density of matter that do survive the inflationary period are so dense that they eventually collapse into galaxy-sized blackholes! Neither the precise way in which the transition to ordinary Hubbel expansion occurs nor the duration of the inflationary period are well determined.

If the inflationary cosmologies can be made to answer each of these issues satisfactorily we may have, as J. Richard Gott III has suggested, a most remarkable model of the universe where an almost infinite number of ‘bubble universes’ each having nearly infinite size, coexist in the same 4-dimensional spacetime; all of these bubble universes having been brought into existence at the same instant of creation. This is less troublesome than one might suspect since, if our universe is actually infinite as the available data suggests, so too was it infinite even at its moment of birth! It is even conceivable that the universe is ‘percolating’ with new bubble universes continually coming into existence. Our entire visible universe, out to the most distant quasar, would be but one infinitessimal patch within one of these bubble regions. Do these other universes have galaxies, stars, planets and living creatures statistically similar to those in our universe? We may never know. These other universes, born of the same paroxicism of Creation as our own, are forever beyond our scrutiny but obviously not our imaginations!

Beyond The Beginning…

Finally, what of the period before Grand Unification? We may surmise that at higher temperatures than the GUT Era, even the supermassive Higgs and Leptoquark bosons become massless and at long last we arrive at a time when the gravitational interaction is united with the weak, electromagnetic and strong forces. Yet, our quest for an understanding of the origins of the universe remains incomplete since gravity has yet to be brought into unity with the remaining forces on a theoretical basis. This last step promises to be not only the most difficult one to take on the long road to unification but also appears to hold the greatest promise for shedding light on some of the most profound mysteries of the physical world. Even now, a handful of theorists around the world are hard at work on a theory called Supergravity which unites the force carriers (photons, gluons, gravitons and the weak interaction bosons) with the particles that they act on (quarks, electrons etc). Supergravity theory also predicts the existence of new particles called photinos and gravitinos. There is even some speculation that the photinos may fill the entire universe and account for the unseen ‘missing’ matter that is necessary to give the universe the critical mass required to make it exactly Euclidean. The gravitinos, on the other hand, prevent calculations involving the exchange of gravitons from giving infinite answers for problems where the answers are known to be perfectly finite. Hitherto, these calculations did not include the affects of the gravitinos.

Perhaps during the next decade, more of the details of the last stage of Unification will be hammered out at which time the entire story of the birth of our universe can be told. This is, indeed, an exciting time to be living through in human history. Will future generations forever envy us our good fortune, to have witnessed in our lifetimes the unfolding of the first comprehensive theory of Existence?

What is Space? Part I

Does Space Have More Than 3 Dimensions?
Written by Sten Odenwald
Copyright (C) 1984 Kalmbach Publishing. Reprinted by permission

The intuitive notion that the universe has three dimensions seems to be an irrefutable fact. After all, we can only move up or down, left or right, in or out. But are these three dimensions all we need to describe nature? What if there aree, more dimensions ? Would they necessarily affect us? And if they didn’t, how could we possibly know about them? Some physicists and mathematicians investigating the beginning of the universe think they have some of the answers to these questions. The universe, they argue, has far more than three, four, or five dimensions. They believe it has eleven! But let’s step back a moment. How do we know that our universe consists of only three spatial dimensions? Let’s take a look at some “proofs.”

On a 2-dimensional piece of paper you can draw an infinite number of polygons.  But when you try this same trick in 3-dimensions you run up against a problem.There are five and only five regular polyhedra. A regular polyhedron is defined as a solid figure whose faces are identical polygons – triangles, squares, and pentagons – and which is constructed so that only two faces meet at each edge. If you were to move from one face to another, you would cross over only one edge. Shortcuts through the inside of the polyhedron that could get you from one face to another are forbidden. Long ago, the mathematician Leonhard Euler demonstrated an important relation between the number of faces (F), edges (E), and corners (C) for every regular polyhedron: C – E + F = 2. For example, a cube has 6 faces, 12 edges, and 8 corners while a dodecahedron has 12 faces, 30 edges, and 20 corners. Run these numbers through Euler’s equation and the resulting answer is always two, the same as with the remaining three polyhedra. Only five solids satisfy this relationship – no more, no less.

Not content to restrict themselves to only three dimensions, mathematicians have generalized Euler’s relationship to higher dimensional spaces and, as you might expect, they’ve come up with some interesting results. In a world with four spatial dimensions, for example, we can construct only six regular solids. One of them – the “hypercube” – is a solid figure in 4-D space bounded by eight cubes, just as a cube is bounded by six square faces. What happens if we add yet another dimension to space? Even the most ambitious geometer living in a 5-D world would only be able to assemble thee regular solids. This means that two of the regular solids we know of – the icosahedron and the dodecahedron – have no partners in a 5-D universe.
For those of you who successfully mastered visualizing a hypercube, try imagining what an “ultracube” looks like. It’s the five- dimensional analog of the cube, but this time it is bounded by one hypercube on each of its 10 faces! In the end, if our familiar world were not three-dimensional, geometers would not have found only five regular polyhedra after 2,500 years of searching. They would have found six (with four spatial dimension,) or perhaps only three (if we lived in a 5-D universe). Instead, we know of only five regular solids. And this suggests that we live in a universe with, at most, three spatial dimensions.

All right, let’s suppose our universe actually consists of four spatial dimensions. What happens? Since relativity tells us that we must also consider time as a dimension, we now have a space-time consisting of five dimensions. A consequence of 5-D space-time is that gravity has freedom to act in ways we may not want it to.

To the best available measurements, gravity follows an inverse square law; that is, the gravitational attraction between two objects rapidly diminishes with increasing distance. For example, if we double the distance between two objects, the force of gravity between them becomes 1/4 as strong; if we triple the distance, the force becomes 1/9 as strong, and so on. A five- dimensional theory of gravity introduces additional mathematical terms to specify how gravity behaves. These terms can have a variety of values, including zero. If they were zero, however, this would be the same as saying that gravity requires only three space dimensions and one time dimension to “give it life.” The fact that the Voyager space- craft could cross billions of miles of space over several years and arrive vithin a few seconds of their predicted times is a beautiful demonstration that we do not need extra-spatial dimensions to describe motions in the Sun’s gravitational field.

From the above geometric and physical arguments, we can conclude (not surprisingly) that space is three-dimensional – on scales ranging from that of everyday objects to at least that of the solar system. If this were not the case, then geometers would have found more than five regular polyhedra and gravity would function very differently than it does – Voyager would not have arrived on time. Okay, so we’ve determined that our physical laws require no more than the three spatial dimensions to describe how the universe works. Or do they? Is there perhaps some other arena in the physical world where multidimensional space would be an asset rather than a liability?

Since the 1920s, physicists have tried numerous approaches to unifying the principal natural interactions: gravity, electromagnetism, and the strong and weak forces in atomic nuclei. Unfortunately, physicists soon realized that general relativity in a four-dimensional space-time does not have enough mathematical “handles” on which to hang the frameworks for the other three forces. Between 1921 and 1927, Theodor Kaluza and Oskar Klein developed the first promising theory combining gravity and electromagnetism. They did this by extending general relativity to five dimensions. For most of us, general relativity is mysterious enough in ordinary four-dimensional space-time. What wonders could lie in store for us with this extended universe?

General relativity in five dimensions gave theoreticians five additional quantities to manipulate beyond the 10 needed to adequately define the gravitational field. Kaluza and Klein noticed that four of the five extra quantities could be identified with the four components needed to define the electromagnetic field. In fact, to the delight of Kaluza and Klein, these four quantities obeyed the same types of equations as those derived by Maxwell in the late 1800s for electromagnetic radiationl Although this was a promising start, the approach never really caught on and was soon buried by the onrush of theoretical work on the quantum theory of electromagnetic force. It was not until work on supergravity theory began in 1975 that Kaluza and Klein’s method drew renewed interest. Its time had finally come.

What do theoreticians hope to gain by stretching general relativity beyond the normal four dimensions of space-time? Perhaps by studying general relativity in a higher-dimensional formulation, we can explain some of the constants needed to describe the natural forces. For instance, why is the proton 1836 times more massive than the electron? Why are there only six types of quarks and leptons? Why are neutrinos massless? Maybe such a theory can give us new rules for calculating the masses of fundamental particles and the ways in which they affect one another. These higher-dimensional relativity theories may also tell us something about the numbers and properties of a mysterious new family of particles – the Higgs bosons – whose existence is predicted by various cosmic unification schemes. (See “The Decay of the False Vacuum,” ASTRONOMY, November 1983.)

These expectations are not just the pipedreams of physicists – they actually seem to develop as natural consequences of certain types of theories studied over the last few years. In 1979, John Taylor at Kings College in London found that some higher- dimensional formalisms can give predictions for the maximum mass of the Higgs bosons (around 76 times that of the proton.) As they now stand, unification theories can do no more than predict the existence of these particles – they cannot provide specific details about their physical characteristics. But theoreticians may be able to pin down some of these details by using extended theories of general relativity. Experimentally, we know of six leptons: the electron, the muon, the tauon, and their three associated neutrinos. The most remarkable prediction of these extended relativity schemes, however, holds that the number of leptons able to exist in a universe is related to the number of dimensions of space-time. In a 6-D space-time, for example, only one lepton – presumably the electron – can exist. In a 10-D space-time, four leptons can exist – still not enough to accommodate the six we observe. In a 12-D space- time, we can account for all six known leptons – but we also acquire two additional leptons that have not yet been detected. Clearly, we would gain much on a fundamental level if we could increase the number of dimensions in our theories just a little bit.

How many additional dimensions do we need to consider in order to account for the elementary particles and forces that we know of today? Apparently we require at least one additional spatial dimension for every distinct “charge” that characterizes how each force couples to matter. For the electromagnetic force, we need two electric charges: positive and negative. For the strong force that binds quarks together to form, among other things, protons and neutrons, we need three “color” charges – red, blue, and green. Finally, we need two “weak” charges to account for the weak nuclear force. if we add a spatial dimension for each of these charges, we end up with a total of seven extra dimensions. The properly extended theory of general relativity we seek is one with an 11 -dimensional space-time, at the very least. Think of it – space alone must have at least 10 dimensions to accomodate all the fields known today.

Of course, these additional dimensions don’t have to be anything like those we already know about. In the context of modern unified field theory, these extra dimensions are, in a sense, internal to the particles themselves – a “private secret,” shared only by particles and the fields that act on them! These dimensions are not physically observable in the same sense as the three spatial dimensions we experience; they’stand in relation to the normal three dimensions of space much like space stands in relation to time.

With today’s veritable renaissance in finding unity among the forces and particles that compose the cosmos, some by methods other than those we have discussed, these new approaches lead us to remarkably similar conclusions. It appears that a four-dimensional space-time is simply not complex enough for physics to operate as it does.

We know that particles called bosons mediate the natural forces. We also know that particles called fermions are affected by these forces. Members of the fermion family go by the familiar names of electron, muon, neutrino, and quark; bosons are the less well known graviton, photon, gluon, and intermediate vector bosons. Grand unification theories developed since 1975 now show these particles to be “flavors” of a more abstract family of superparticies – just as the muon is another type of electron. This is an expression of a new kind of cosmic symmetry – dubbed supersymmetry, because it is all-encompassing. Not only does it include the force-carrying bosons, but it also includes the particles on which these forces act. There also exists a corresponding force to help nature maintain supersymmetry during the various interactions. It’s called supergravity. Supersymmetry theory introduces two new types of fundamental particles – gravitinos and photinos. The gravitino has the remarkable property of mathematically moderating the strength, of various kinds of interactions involving the exchange of gravitons. The photino, cousin of the photon, may help account for the “missing mass” in the universe.

Supersymmetry theory is actually a complex of eight different theories, stacked atop one another like the rungs of a ladder. The higher the rung, the larger is its complement of allowed fermion and boson particle states. The “roomiest” theory of all seems to be SO(8), (pronounced ess-oh-eight), which can hold 99 different kinds of bosons and 64 different kinds of fermions. But SO(8) outdoes its subordinate, SO(7), by only one extra dimension and one additional particle state. Since SO(8) is identical to SO(7) in all its essential features, we’ll discuss SO(7) instead. However, we know of far more than the 162 types of particles that SO(7) can accommodate, and many of the predicted types have never been observed (like the massless gravitino). SO(7) requires seven internal dimensions in addition to the four we recognize – time and the three “every day” spatial dimensions. If SO(7) at all mirrors reality, then our universe must have at least 11 dimensions! Unfortunately, it has been demonstrated by W. Nahm at the European Center for Nuclear Research in Geneva, Switzerland that supersymmetry theories for space-times with more than 11 dimensions are theoretically impossible. SO(7) evidently has the largest number of spatial dimensions possible, but it still doesn’t have enough room to accommodate all known types of particles.

It is unclear where these various avenues of research lead. Perhaps nowhere. There is certainly ample historical precedent for ideas that were later abandoned because they turned out to be conceptual dead-ends. Yet what if they turn out to be correct at some level? Did our universe begin its life as some kind of 11-dimensional “object” which then crystallized into our four- dimensional cosmos?

Although these internal dimensions may not have much to do with the real world at the present time, this may not always have been the case. E. Cremmer and J. Scherk of I’Ecole Normale Superieure in Paris have shown that just as the universe went through phase transitions in its early history when the forces of nature became distinguishable, the universe may also have gone through a phase transition when mensionality changed. Presumably matter has something like four external dimensions (the ones we encounter every day) and something like seven internal dimensions. Fortunately for us, these seven extra dimensions don’t reach out into the larger 4-D realm where we live. If they did, a simple walk through the park might become a veritable obstacle course, littered with wormholes in space and who knows what else!

Alan Chocos and Steven Detweiler of Yale University have considered the evolution of a universe that starts out being five- dimensional. They discovered that while the universe eventually does evolve to a state where three of the four spatial dimensions expand to become our world at large, the extra fourth spatial dimension shrinks to a size of 10^-31 centimeter by the present time. The fifth dimension to the universe has all but vanished and is 20 powers of 10 – 100 billion billion times – smaller than the size of a proton. Although the universe appears four- dimensional in space-time, this perception is accidental due to our large size compared to the scale of the other dimensions. Most of us think of a dimension as extending all the way to infinity, but this isn’t the full story. For example, if our universe is really destined to re-collapse in the distant future, the three- dimensional space we know today is actually limited itself – it will eventually possess a maximum, finite size. It just so happens that the physical size of human beings forces us to view these three spatial dimensions as infinitely large.

It is not too hard to reconcile ourselves to the notion that the fifth (or sixth, or eleventh) dimension could be smaller than an atomic nucleus – indeed, we can probably be thankful that this is the case.