Tag Archives: space

Quantum Gravity…Oh my!

So here’s the big problem.

Right now, physicists have a detailed mathematical model for how the fundamental forces in nature work: electromagnetism, and the strong and weak nuclear forces. Added to this is a detailed list of the fundamental particles in nature like the electron, the quarks, photons, neutrinos and others. Called the Standard Model, it has been extensively verified and found to be an amazingly accurate way to describe nearly everything we see in the physical world. It explains why some particles have mass and others do not. It describes exactly how forces are generated by particles and transmitted across space. Experimenters at the CERN Large Hadron Collider are literally pulling out their hair to find errors or deficiencies in the Standard Model that go against the calculated predictions, but have been unable to turn up anything yet. They call this the search for New Physics.

Along side this accurate model for the physical forces and particles in our universe, we have general relativity and its description of gravitational fields and spacetime. GR provides no explanation for how this field is generated by matter and energy. It also provides no description for the quantum structure of matter and forces in the Standard Model. GR and the Standard Model speak two very different languages, and describe two very different physical arenas. For decades, physicists have tried to find a way to bring these two great theories together, and the results have been promising but untestable. This description of gravitational fields that involves the same principles as the Standard Model has come to be called Quantum Gravity.

The many ideas that have been proposed for Quantum Gravity are all deeply mathematical, and only touch upon our experimental world very lightly. You may have tried to read books on this subject written by the practitioners, but like me you will have become frustrated by the math and language this community has developed over the years to describe what they have discovered.

The problem faced by Quantum Gravity is that gravitational fields only seem to display their quantum features at the so-called Planck Scale of 10^-33 centimeters and  10^-43 seconds. I cant write this blog using scientific notation, so I am using the shorthand that 10^3 means 1000 and 10^8 means 100 million. Similarly, 10^-3 means 0.001 and so on. Anyway, the Planck scale  also corresponds to an energy of 10^19 GeV or 10 billion billion GeV, which is an energy 1000 trillion times higher than current particle accelerators can reach.

There is no known technology that can reach the scales where these effects can be measured in order to test these theories. Even the concept of measurement itself breaks down! This happens because the very particles (photons) you try to use to study physics at the Planck scale carry so much energy  they turn into quantum black holes and are unable to tell you what they saw or detected!

One approach to QG is called Loop Quantum Gravity.  Like relativity, it assumes that the gravitational field is all there is, and that space and time become grainy or ‘quantized’ near the Planck Scale. The space and time we know and can experience in-the-large is formed from individual pieces that come together in huge numbers to form the appearance of a nearly-continuous and smooth gravitational field.

The problem is that you cannot visualize what is going on at this scale because it is represented in the mathematics, not by nuggets of space and time, but by more abstract mathematical objects called loops and spin networks. The artist rendition above is just that.

So here, as for Feynman Diagrams, we have a mathematical picture that represents a process, but the picture is symbolic and not photographic. The biggest problem, however, is that although it is a quantum theory for gravity that works, Loop Quantum Gravity does not include any of the Standard Model particles. It represents a quantum theory for a gravitational field (a universe of space and time) with no matter in it!

In other words, it describes the cake but not the frosting.

The second approach is string theory. This theory assumes there is already some kind of background space and time through which another mathematical construct called a string, moves. Strings that form closed loops can vibrate, and each pattern of vibrations represents a different type of fundamental particle. To make string theory work, the strings have to exist in 10 dimensions, and most of these are wrapped up into closed balls of geometry called Calabi-Yau spaces. Each of these spaces has its own geometry within which the strings vibrate. This means there can be millions of different ‘solutions’ to the string theory equations: each a separate universe with its own specific type of Calabi-Yau subspace that leads to a specific set of fundamental particles and forces. The problem is that string theory violates general relativity by requiring a background space!

In other words, it describes the frosting but not the cake!

One solution proposed by physicist Lee Smolin is that Loop Quantum Gravity is the foundation for creating the strings in string theory. If you looked at one of these strings at high magnification, its macaroni-like surface would turn into a bunch of loops knitted together, perhaps like a Medieval chainmail suit of armor. The problem is that Loop Quantum Gravity does not require a gravitational field with more than four dimensions ( 3 of space and one of time) while strings require ten or even eleven. Something is still not right, and right now, no one really knows how to fix this. Lacking actual hard data, we don’t even know if either of these theories is closer to reality!

What this hybrid solution tries to do is find aspects of the cake that can be re-interpreted as particles in the frosting!

This work is still going on, but there are a few things that have been learned along the way about the nature of space itself. At our scale, it looks like a continuous gravitational field criss-crossed by the worldlines of atoms, stars and galaxies. This is how it looks even at the atomic scale, because now you get to add-in the worldlines of innumerable ‘virtual particles’ that make up the various forces in the Standard Model.  But as we zoom down to the Planck Scale, space and spacetime stop being smooth like a piece of paper, and start to break up into something else, which we think reveals the grainy nature of gravity as a field composed of innumerable gravitons buzzing about.

But what these fragmentary elements of space and time ‘look’ like is impossible to say. All we have are mathematical tools to describe them, and like our attempts at describing the electron, they lead to a world of pure abstraction that cannot be directly observed.

If you want to learn a bit more about the nature of space, consider reading my short booklet ‘Exploring Quantum Space‘ available at amazon.com. It describes the amazing history of our learning about space from ancient Greek ‘common sense’ ideas, to the highlights of mind-numbing modern quantum theory.

Check back here on Thursday, December 22 for the last blog in this series!

What IS space?

One thing that is true about physics is that it involves a lot of mathematics. What this means is that we often use the mathematics to help us visualize what is going on in the world. But like I said in an earlier blog, this ‘vision thing’ in math can sometimes let you mistake the model for the real thing, like the case of the electron. The same problem emerges when we try to understand an invisible  thing like space.

The greatest discovery about space  was made by Einstein just before 1915 as he was struggling to turn his special theory of relativity into something more comprehensive.

Special relativity was his theory of space and time that described how various observers would see a consistent world despite their uniform motion at high speeds. This theory alone revolutionized physics, and has been the main-stay of modern quantum mechanics, as well as the designs of powerful accelerators that successfully and accurately push particles to nearly the speed of light. The problem was that special relativity did not include a natural place for accelerated motion, especially in gravitational fields, which are of course very common in the universe.

Geometrically, special relativity only works when worldlines are perfectly straight, and  form lines within a perfectly flat, 4-dimensional spacetime (a mathematical arena where 3 dimensions of space are combined with one dimension of time). But accelerated motion causes worldlines to be curved, and you cannot magically make the curves go straight again and keep the spacetime geometrically flat just by finding another coordinate system.

Special relativity, however, promised that so long as motion is at constant speed and worldlines are straight, two different observers (coordinate systems) would agree about what they are seeing and measuring by using the mathematics of special relativity. With curved worldlines and acceleration, the equations of special relativity, called the Lorentz Transformations, would not work as they were. Einstein was, shall we say, annoyed by this because clearly there should be some mathematical process that would allow the two accelerated observers to again see ( or calculate) consistent physical phenomena.

He began his mathematical journey to fix this problem by writing his relativity equations in a way that was coordinate independent using the techniques of tensor analysis. But he soon found himself frustrated by what he needed in order to accomplish this mathematical miracle, versus his knowledge of advanced analytic geometry in four dimensions. So he went to his classmate and math wiz, Marcel Grossman, who immediately recognized that Einstein’s mathematical needs were just an awkward way of stating certain properties of non-Euclidean geometry developed by Georg Riemann and others in the mid-to-late 1800s.

This was the missing-math that Einstein needed, who being a quick learner, mastered this new language and applied it to relativity. After an intense year of study, and some trial-and-error mathematical efforts, he published his complete Theory of General Relativity in November 1915. Just like the concept of spacetime did away with space and time as independent ideas in special relativity, his new theory made an even bigger, revolutionary, discovery.

It was still a theory of the geometry of worldlines that he was proposing, but now the geometric properties of these worldlines was controlled by a specific mathematical term called the metric tensor. This mathematical object was fundamental to all geometry as Grossman had showed him, and allowed you to calculate distances between points in space. It also defined what a ‘straight line’ meant, as well as how curved the space was. Amazingly, when you translated all this geometric talk into the hard, cold reality of physics in 4-dimensions, this metric tensor turned into the gravitational field through which the worldline of a particle was defined as the straightest-possible path.

An interesting factoid, indeed, but why is it so revolutionary?

All other fields in physics (e.g like the electromagnetic field) are defined by some quantity, call it A, that is specified at each coordinate point in space and time: A(x,y,z,t). If you take-away the field, the coordinate grid remains intact. But with the gravitational field, there is no background coordinate grid to define its intensity, instead, the gravitational field provides its own coordinate grid because it is identical to the metric tensor!!

This is why Einstein and physicists say that gravity is not a force like the others we know about, but instead it is a statement about the shape of the geometry of spacetime through which particles move. (Actually, particles do not move through spacetime. Their histories from start to finish simply exist all at once like a line drawn on a piece of paper!)

So, imagine a cake with frosting on it. The frosting represents the various fields in space, and you can locate where they are and how much frosting is on the cake from place to place. But the bulk of the cake, which is supporting the frosting and telling you that ‘this is the top, center, side, etc of the cake’ is what supports the frosting. Take away the cake, and the frosting is unsupported, and can’t even be defined in the first place. Similarly, take away the gravitational field, symbolized by Einstein’s metric tensor, and spacetime actually disappears!

Amazingly, Einstein’s equations say that although matter and energy produce gravitational fields, you can have situations where there is no matter and energy and spacetime still doesn’t vanish! These vacuum solutions are real head-scratchers when physicists try to figure out how to combine quantum mechanics, our premier theory of matter, with general relativity: our premier theory of gravity and spacetime. These vacuum solutions represent gravitational fields in their purest form, and are the starting point for learning how to describe the quantum properties of gravitational fields. They are also important to the existence of gravity waves, which move from place to place as waves in the empty spacetime between the objects producing them.

But wait a minute. Einstein originally said that ‘space’ isn’t actually a real thing. Now we have general relativity, which seems to be bringing space (actually spacetime) back as something significant in its own right as an aspect of the gravitational field.

What gives?

To see how some physicists resolve these issues, we have to delve into what is called quantum gravity theory, and this finally gets us back to some of my earlier blogs about the nature of space, and why I started this blog series!

 

Check back here on Wednesday, December 21 for the last installment on this series about space!

Is Infinity Real?

In the daytime, you are surrounded by trees, buildings and the all-too-familiar accoutrements of Nature, to which by evolution we were designed to appreciate and be familiar. But at night, we see an unimaginably different view: The dark, starry night sky, with no sense of perspective or depth. It is easy to understand how The Ancients thought it a celestial ceiling with pinpoint lights arrayed in noteworthy patterns. Many millennia of campfires were spent trying to figure it out.

We are stuck in the middle ground between two vast scales that stretch before us and within us. Both, we are told, lead to the infinitely-large and the infinitely-small. But is this really true?

Astronomically, we can detect objects that emerged from the Big Bang nearly 14 billion years ago, which means their light-travel distance from us is 14 billion light years or 13,000,000,000,000,000,000,000,000,000 centimeters. This is, admittedly, a big number but it is not infinitely-large.

In the microcosm, we have probed the structure of electrons to a scale of 0.000000000000000000001 centimeters and found no signs of any smaller distance yet. So again, there is no sign that we have reached anything like an infinitely-small limit to Nature either.

When it comes right down to it, the only evidence we have for the universe being infinitely large (or other aspects of it being infinitely small) is in the mathematics and geometry we use to describe it. Given that infinity is the largest number you can count to, it is pretty obvious that even the scale of our visible universe of 13,000,000,000,000,000,000,000,000,000 centimeters falls woefully short of being even a relatively stupendous number by comparison to infinity.

Infinity is as old as the Ancient Greeks. But even Aristotle (384 – 322 BCE) would only allow the integers (1,2,3,…) to be potentially infinite, but not actually infinite, in quantity. Since then, infinity or its cousin eternity, have become a part of our literary and religious vernacular when we mention something really, really, really….. big or old! Through literary and philosophical repetition, we have become comfortable with this idea in a way that is simply not justifiable.

Mathematics can define infinity very precisely, and even the mathematician Cantor (1845 – 1918) was able to classify ‘transfinite numbers’ as being either representing countable infinities or uncountable infinities. To the extent that mathematics is also used in physics, we inherit infinity as the limit to many of our calculations and models of the physical world. But the problem is that our world is only able to offer us the concept of something being very, very, very… big, like the example of the visible universe above.

If you take a sphere a foot across and place an ant on it, it crawls around and with a bit of surveying it can tell you the shape is a sphere with a finite closed surface. But now take this sphere and blow it up so that it is 1 million miles across. The ant now looks across its surface and sees something that looks like an infinite plane. Its geometry is as flat as a sheet of paper on a table.

In astronomy we have the same problem.

We make calculations and measurements within the 28 billion light years that spans our visible universe and conclude that the geometry of the universe is flat, and so geometrically it seems infinite, but the only thing the measurements can actually verify is that the universe is very, very, very large and LOOKS like its geometry is that of an infinite, flat, 3-dimensional space. But modern Big Bang cosmology also says that what we are seeing within our visible universe is only a portion of a larger thing that emerged from the Big Bang and ‘inflated’ to enormous size in the first microseconds.  If you identify our visible universe out to 14 billion light years as the size of the period at the end of this sentence, that larger thing predicted by inflation may be millions of miles across at the same scale. This is very, very big, but again it is not infinite!

Going the other way, the current best theoretical ideas about the structure of the physical world seems to suggest that at some point near a so-called Planck scale of 0.0000000000000000000000000000000015 centimeters we literally ‘run out of space’. This mathematical conclusion seems to be the result of combining the two great pillars of all physical science, quantum mechanics and general relativity, into a single ‘unified’ theory.  The mathematics suggests that, rather than being able to probe the nature of matter and space at still-smaller scales, the entire edifice of energy, space, time and matter undergoes a dramatic and final change into something vastly different than anything we have ever experienced: elements that are beyond space and time themselves.  These ideas are captured in theories such as Loop Quantum Gravity and String Theory, but frankly we are still at a very early stage in understanding what this all means. Even more challenging is that we have no obvious way to make any measurements that would directly test whether physical reality simply comes to an end at these scales or not.

So on the cosmological scene, we can convincingly say we have no evidence that anything as large as ‘infinity’ exists because it is literally beyond our 14 billion light-year horizon of detection. The universe is simply not old enough for us to sample such an imponderably large realm. Advances in Big Bang cosmology can only propose that we live in an incomprehensively alien ‘multiverse’ or that we inhabit one miniscule dot in a vastly larger cosmos, which our equations extrapolate as infinity. Meanwhile, the world of the quantum hints that no infinitely-small structures exist in the universe, not even what we like to call space itself can be indefinitely sub-divided below the Planck scale.

In the end, it seems that infinity is a purely  mathematical ideal that can be classified by Cantor’s transfinite numbers manipulated symbolically, and thought about philosophically, but is never actually found among the objects that inhabit our physical world.

Now let’s go back to the issue of space after the relativity revolution and try to make sense of where we stand now!

Check back here on Monday, December 19 for the next installment!

Is Space Real?

I take a walk to the store and can’t help but feel I am moving through something that is more than the atmosphere that rushes by my face as I go. The air itself is contained within the boundaries of the space through which I pass. If I were an astronaut in the vacuum of outer space, I would still have the sense that my motion was through a pre-existing, empty framework of 3-dimensions. Even if I were blind and confined to a wheelchair, I could still have the impression through muscular exertion that I was moving through space to get from my kitchen to my living room ‘over there’. But what is space as a physical thing? Of all the phenomena, forces and particles we study, each is something concrete though generally invisible: a field; a wave; a particle. But space, itself, seems to be none of these. WTF!

Spider web covered with dew drops

Way back in the early 1700s, Sir Isaac Newton proposed that space was an ineffable, eternal framework through which matter passed. It had an absolute and immutable nature. Its geometry pre-existed the matter that occupied it and was not the least bit affected by matter. A clever set of experiments in the 20th century finally demonstrated rather conclusively that there is no pre-existing Newtonian space or geometry ‘beneath’ our physical world. There is no absolute framework of coordinates within which our world is embedded. What had happened was that Albert Einstein developed a new way of thinking about space that essentially denied its existence!

Albert Einstein’s relativity revolution completely overturned our technical understanding of space and showed that the entire concept of dimensional space was something of a myth. In his famous quote he stressed that We entirely shun the vague word ‘space’ of which we must honestly acknowledge we cannot form the slightest conception. In the relativistic world we live in, space has no independent existence. “…[prior-geometry] is built on the a priori, Euclidean [space], the belief in which amounts to something like a superstition“. So what could possibly be a better way of thinking about space than the enormously compelling idea that each of us carries around in our brains, that space is some kind of stage upon which we move?

To understand what Einstein was getting at, you have to completely do away with the idea that space ‘is there’ and we move upon it or through it. Instead, relativity is all about the geometry created by the histories (worldlines) of particles as they move through time. The only real ‘thing’ is the collection of events along each particle’s history. If enough particles are involved, the histories are so numerous they seem like a continuous space. But it is the properties of the events along each history that determine the over-all geometry of the whole shebang and the property we call ‘dimension’, not the other way around.

This figure is an example where the wires (analogous to worldlines) are defining the shape and contours of a dimensional shape. There is nothing about the background (black) space that determines how they bend and curve. In fact, with a bit of mathematics you could specify everything you need to know about the surface of this shape and from the mathematics tell what the shape is, and how many dimensions are required to specify it!

Princeton University physicist Robert Dicke expressed it this way, “The collision between two particles can be used as a definition of a point in [space]…If particles were present in large numbers…collisions could be so numerous as to define an almost continuous trajectory…The empty background of space, of which ones knowledge is only subjective, imposes no dynamical conditions on matter.”

What this means is that so long as a point in space is not occupied by some physical event such as the interaction point of a photon and an electron, it has no effect on a physical process ( a worldline) and is not even observable. It is a mathematical ‘ghost’ that has no effect on matter at all. The interstitial space between the events is simply not there so far as the physical world based upon worldlines is concerned. It is not detectable even by the most sophisticated technology, or any inventions to come. It does not even supply something as basic as the ‘dimension’ for the physical world!

We should also be mindful of another comment by Einstein that “…time and space are modes by which we think and not conditions in which we live“. They are free creations of the human mind, to use one of Einstein’s own expressions. By the way, the 18th century philosopher Immanuel Kant also called the idea of ‘space’ an example of a priori knowledge that we are born with to sort out the world, but it is not necessarily a real aspect of the world outside our senses.

Like a spider web, individual and numerous events along a worldline define the worldline’s shape, yet like the spider web, this web can be thought of as embedded in a larger domain of mathematically-possible events that could represent physical events…but don’t. The distinction between these two kinds of points is what Einstein’s revolutionary idea of relativity provided physicists, and is the mainstay of all successful physical theories since the 1920s. Without it, your GPS-enabled cell phones would not work!

So what are these events? Simply put, according to Physicist Lee Smolin, they are exchanges of information, which are also the interaction points between one particle’s worldline and another particle’s world line. If you think at the atomic level, each time a particle of light interacts with (collides or is emitted by) an electron it generates an event. These events are so numerous the electron’s worldline looks like a continuous line with no gaps between the events. So the shape of one worldline, what we call its history, is a product of innumerable interactions over time with the worldlines of all other objects (photons etc) to which it can be in cause-and-effect contact.

Even though this new idea of space being a myth has gained enormous validity among physicists over the last century, and I can easily speak the language of relativity to describe it, personally, my mind has a hard time really understanding it all. I also use the mathematical theory of quantum mechanics to make phenomenally accurate predictions, but no Physicist really understands why it works, or what it really means.

Next time I want to examine how the history of a particle is more important than the concept of space in Einstein’s relativity, and how this explains the seeming rigidity of the world you perceive and operate within.

Check back here on Thursday, December 15 for the next installment!