|HPS 0410||Einstein for Everyone|
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John D. Norton
Department of History and Philosophy of Science
University of Pittsburgh
Background reading: J. P. Mc Evoy and O. Zarate, Introducing Stephen Hawking. Totem. pp. 46-105.
Which solutions of Einstein's gravitational field equations can accommodate a universe with a Hubble expansion? Or, more figuratively, which pages in Einstein's great book might belong to our universe?
|It may seem an overwhelming task to pore through the pages of Einstein's great book to find those that might be our universe. However it is not so. We need to pick just two properties of the spacetime to narrow down our choices to a family of spacetimes that form the basis of modern, relativistic cosmology.||The situation is not so different from someone trying to identify a small country in a huge atlas of the world. The atlas may be huge. But, if it is well made, one needs only a few key words and a good index to arrive rapidly at the page of interest.|
The two conditions are that the ordinary, three dimensional spaces of our cosmos are homogeneous and isotropic; and that space is filled uniformly with matter. They suffice to identify a class of solutions of Einstein's equations known as the Friedmann-Robertson-Walker spacetimes, or, as it is sometimes called, the Friedmann-Lemaître-Robertson-Walker spacetimes.
Recall how we first built a spacetime many chapters ago. We took all of space at many instants of time and stacked these spaces up into a fourth, temporal dimension to generate spacetime.
|We will stack these spaces again to build the spacetimes of relativistic cosmology. When we first stacked these spaces to build a Minkowski spacetime of special relativity, we assumed tacitly with Einstein and Minkowski that space was Euclidean. No other possibility seemed reasonable when special relativity was born. Now we know better. The spaces we will stack up do not have to be Euclidean. We will leave open the possible geometries.||That a spacetime can be built up this way seems obvious and natural. However not all spacetimes can be built up this way. We have already seen one. A Goedel universe cannot be sliced up nicely into spaces that evolve into each other over time. Reversing the failure, we cannot build a Goedel spacetime by stacking up snapshots of space taken at different moments of time.|
What geometries can the spaces have? The observations of our cosmos tell us that, on the largest scale, space at any instant of time is homogeneous and isotropic. So we ask that the geometries of the spaces we stack up are homogeneous and isotropic . That means that, in each case, the geometry of the space is the same in every place (homogeneous) and is the same in all directions (isotropic).
This condition that space is homogeneous and isotropic on the largest scale has been called the "cosmological principle." That strikes me as a little dangerous. Naming the condition the cosmological principle does no harm, of course, as long as we realize that it is just a name. However, when the term "principle" is used, it is easy to get the impression that the condition is somehow unchallengeable. That is risky. Whether the universe is roughly homogeneous and isotropic is something that should be determined by observation. It should not be legislated in advance.
The condition that the space is homogeneous and isotropic restricts the geometry to three general possibilities. Such a space must have constant spatial curvature. We know from earlier that the three possibilities are:
A space of one of these three types will be the instantaneous snapshots that comprise the "now" of the cosmology.
Each of these snapshots of space will be filled with a uniform matter distribution. Its composition is not fixed. However it is sufficient to distinguish two general types of matter to pick out the spacetimes.
The first type is ordinary matter, such as comprises planets and stars. We have seen that, on the larger scale, this matter is clustered into stars and then galaxies. On the largest scale, these galaxies fill space just as the molecules of air fill the space of a room. In both cases, if we work at a sufficiently large scale, we can ignore the granularity of individual molecules or galaxies. In both cases, matter behaves like a fluid. Sometimes, cosmologists label the galactic matter as "dust." To understand why, think of a dust so fine that flows like a fluid.
The second type of matter is radiant matter. It is the type of matter Penzias and Wilson found to comprise the cosmic background radiation. We need to make few assumptions about this second form of matter. All we need is that it is radiant and distributed homogeneously and isotropically in space.
We make no assumption about how much of the cosmic matter is ordinary "dust" and how much is radiant. How the relative proportions evolve will be recovered as a consequence of Einstein's theory.
Once we have specified these two conditions, we have narrowed the possibilities to the Friedmann-Robertson-Walker spacetimes of modern relativistic cosmology. The remaining cosmic properties are now determined through them by Einstein's gravitational field equations.
|The most important property is that the spacetime is dynamic. That is, the spaces of the cosmology cannot remain unchanged. They are either expanding or contracting. As they do this, they carry matter with them. The first
case of expansion is the one that interests us most, since it is what we
observe in the expansion of the galaxies. As space expands and matter is carried with it, the distance between the galaxies increases. The figure shows
the worldlines of the galaxies piercing through the growing spatial slices, the snapshots of space at different moments of time.
In the early stages of this expansion, shortly after the big bang, most of the matter is in the form of extremely hot radiation. The universe is radiation dominated. As the expansion continues, the radiant matter cools and dilutes faster than the ordinary "dust" matter. In latter phases, such as our cosmos is in now, most of the matter is in the form of dust.
|To keep things simple, it is assumed here that the cosmological constant λ is zero. A non-zero cosmological constant opens more possibilities, including Einstein's 1917 static universe.|
The expansion affects the geometry of space as well. As time passes, space at one time expands and evolves into space at a later time. In the process, the curvature of space, if it has any,
These spacetimes come equipped with a unique notion of cosmic time. It appears as the uniqueness of the slicing up of the spacetimes into ordinary three dimensional spaces. Cosmic time is measured by the proper time elapsed along the world lines of the galaxies. In this way, these spacetimes differ from the Minkowski spacetime of special relativity.
In a Minkowski spacetime, we are free to slice up the spacetime in many equivalent ways into ordinary three dimensional spaces. This was the geometric expression of the relativity of simultaneity. It derived directly from the equivalence of all inertial frames of reference and the fact that there is no absolute state of rest in special relativity.
This feature is lost in the Friedmann-Robertson-Walker spacetimes. There is now a unique way to slice up the spacetimes into ordinary three dimensional spaces. It is shown in the figure above. One might worry that this somehow contradicts special relativity. It does not. Special relativity is the theory that describes Minkowski spacetimes. The Friedmann-Robertson-Walker spacetimes are not Minkowski spacetimes and will not conform to the requirements of special relativity
On might still worry that this unique way of slicing up the spacetimes introduces something undesirable, perhaps akin to the absolute state of rest of the nineteenth ether. That does not happen. The problem with the nineteenth century ether was that its absolute state of rest was unobservable. One inertial state of motion out of infinitely many, we were assured, corresponded to absolute rest, but no observation could tell us which that one was.
There is no corresponding problem in these cosmological spacetimes. There is a unique state of rest and it is observable. It is the frame of reference in which cosmic matter is distributed uniformly. If we are at rest in that frame of reference, we will observe cosmic matter to be distributed uniformly around us. If we are moving relative to this cosmic rest frame, we will observe anisotropies. The situation is quite like what happens when we move through calm air. We know that we are moving through the air in a definite direction because we feel the pressure of the air on the leading side.
As it happens, we are not precisely at rest according to this cosmic frame. We are moving, slowly by cosmic standards, at 371 meters per second towards the constellation of Leo. We know this by measurements in the cosmic background radiation. They reveal a slight Doppler shift corresponding to our motion.
The most interesting feature of Friedmann-Robertson-Walker spacetimes is what happens when we project the worldlines of the galaxies into the past. The galaxies get closer and closer. Eventually, they converge onto a state of infinite curvature and density. This is the initial state--the so-called "big bang," shown at the bottom of the figure above.
It is easy to misunderstand the nature of the big bang and the expansion of the universe.
The popular image called to mind by the name big bang is something like this. There is a huge empty space, with an infinitely dense nugget of matter containing all future matter of the universe. At the moment of the big bang, This nugget explodes. Fragments of this primeval nugget are scattered into space, progressively filling it with an expanding cloud of matter. This is NOT the modern big bang model.
Rather the expansion is the expansion of space itself. The most helpful picture is of the rubber surface of a balloon expanding. The galaxies are like dots drawn on the surface. They move as the rubber sheet stretches. The galaxies fly apart because space expands. At any instant, space is always full of matter; there is no island of fragments expanding into an empty space.
Here's how one fixed volume of space will look over the course of cosmic evolution.
If we now project back to the big bang, we project back to a time at which all matter and space were somehow compressed into a state of infinite density. Einstein's gravitational field equations tell us the matter density equals the summed spacetime curvature. So, if the matter density is infinite, the curvature of spacetime has become infinite as well.
That last statement cannot be literally correct. According to Einstein's general theory of relativity, spacetime at every event has definite curvature. If that curvature is everywhere infinite, we define no spacetime at all. If we try to imagine the time of the big bang itself as one of the times of the cosmology, we are saying that there is a time at which spacetime is not properly defined. So there can be no time in the cosmology corresponding to the big bang. We describe the big bang as a "singularity," a breakdown in the laws that govern space and time.
singularity, roughly speaking, designates a point in a mathematical
structure where a quantity fails to be well defined, even though the quantity
is well defined at all neighboring points. The simplest and best known example
arises with the inverse function, 1/x. As long as x is non-zero, 1/x is well
x = 10, 5, 1, 0.5, 0.1, 0.01, ...,
1/x = 0.1, 0.2, 1, 2, 10, 100, ...
For negative values
x = -10, -5, -1, -0.5, -0.1, -0.01...,
1/x is -0.1, -0.2, -1, -2, -10, -100, ...
The system has a singularity at x=0, for then 1/x = 1/0 and, as we all learn in our arithmetic classes, "you cannot divide by zero." There is a temptation to say that 1/0 is "infinity." But that is dangerous. As we have just seen, if we approach x=0 from positive values of x, the inverse 1/x grows without limit towards +infinity (i.e. "plus infinity"). If we approach x=0 from negative values of x, the inverse 1/x grows negatively without limit towards -infinity (i.e. "minus infinity"). If we insist on giving 1/0 a value, which do we give? "Plus-infinity" or "minus-infinity"? The safer course is just to say that we have a singularity at x=0 and not try to give it any value.
What we can say is this. The universe has an age or time--its age after the big bang. The spacetime of the universe exists for every age greater than zero: 1 million years, one hundred years, one second, one half second, one tenth second, and so on. No matter how small we make the age, there is a corresponding spacetime, as long as the age is greater than zero. But nothing in general relativity--no spacetime at all--corresponds to the zero age.
This moment of zero age is a fictitious moment in the history of the universe. In that regard, it is like the fictitious point "at infinity" on the horizon where parallel lines meet. Of course well all know that there is no such point, although we see it drawn routinely in perspective drawings.
We can now return to the red shift that figures in the Hubble expansion and give a more precise account of its origin. It is not a traditional Doppler shift, but something more subtle. A distant galaxy emits light towards us. The light waves with their crests are carried by space towards us. For a distant galaxy, it can take a very long time for the light to reach us. During that time, the cosmic expansion of space proceeds. The effect is that the waves of the light signal get stretched with space. So the wavelength of the light increases and its frequency decreases. It becomes red shifted.
To get a sense of the process, imagine a column of ants setting off to walk across a rubber sheet. They may enter the sheet at a rate of one ant per second. If the rubber sheet is stretched while the ants walk, each ant will need to go further to get to the other side than the one before. So the ants will arrive less frequently at the other side than the original rate of one ant per second.
What is the overall dynamics of spacetime? Einstein's gravitational field equations applied to the Friedmann-Robertson-Walker spacetimes give us three possibilities, cataloged below as I, II or III. (Recall that we have the special case of λ is zero.)
What decides between them is the density of matter. The
so-called "critical density" of matter is the
deciding value. It is a minute average density of 10-29grams per
cubic centimeter. Our cosmology will be one of the three shown in the table
below according to whether the actual average density of matter in our universe
is greater than, equal to or less than this critical density.
|Average mass density||Greater than critical||Critical||Less than critical|
|Geometry of space||Spherical
|Dynamics||Expands and collapses to big crunch||Expands indefinitely||Expands indefinitely|
The table gives the broad features. In cases I and III, space is curved. The scale factor R--the radius of curvature of the space--determines the extent of curvature. (The radius of curvature of a three dimensional space is the three-dimensional analog of the radius of a two-dimensional sphere.) The value of R differs greatly according to the particular matter density at hand. However a rough estimate is this:
So by this estimate the scale factor is roughly 14 billion light years. This value only obtains exactly for special cases. In cosmologies I, it obtains exactly if the average matter density is twice the critical.
We can also get a sense of the dynamics by plotting how the scale factor R changes with time in typical examples of the three cosmologies. In the case of cosmologies II with Euclidean geometry, the scale factor R is simply set to be the distance between two conveniently placed galaxies. As the cosmic expansion proceeds, R grows in response.
In general there are no simple formulae for these curves. One case proves to be simple. In Cosmologies II, if all the matter is what is called "dust" in the jargon (i.e. ordinary matter like our earth), then R increases in direct proportion with (time)2/3. Or in that cosmology, if all the matter is radiation, R increases in direct proportion with (time)1/2.
At first the dynamics seems arbitrary. Why should the different universes have the properties they do? Why, for example, should a universe with greater mass density only have a big crunch? And why with lesser mass density, will the expansion continue indefinitely? We can makes some sense of this with an analogy from Newtonian theory.
There is a reason Newtonian theory can tell us something. Recall that general relativity turns back into Newtonian theory as long as we consider ordinary conditions: nothing moves quickly, there are no strong gravitational fields and--most important here--we consider small distances, not cosmic distances.
So it turns out that a tiny chunk of the cosmic fluid of a Friedmann-Robertson-Walker spacetime is governed by Newtonian principles. The easiest way to see those principles in action is to consider a closely analogous system in Newtonian theory.
Imagine that we have a bomb in space that explodes. It will spread debris into space. Each fragment in the debris will attract all the others according to Newton's inverse square law of gravity. The ultimate fate of the debris cloud depends on the balancing of the initial magnitude of the explosion with the strength of the gravitational attraction within the debris cloud.
If there is a greater amount of matter in the original lump, the explosion will produce a denser cloud of debris. Its internal forces of gravitational attraction will be strong enough to slow and halt the initial outward motion of the explosion and draw the fragments back together, bringing about a collapse. It corresponds to the dynamics of cosmology I; there is a big bang and a big crunch.
If there is a lesser amount of matter in the original lump, the explosion will produce a more dilute debris cloud whose internal forces of attraction will not be sufficient to halt the initial outward motion of the blast. That outward motion will continue indefinitely. It corresponds to cosmology III; there is a big bang and no big crunch.
We could imagine an intermediate case in which the explosion is just energetic enough to fling the debris out of the reach of the gravitational forces; any weakening of the explosion would be too weak to prevent collapse. This corresponds to the intermediate case of cosmology II.
The Newtonian analogy is useful in so far as it gives us a nice picture for the dynamics. But it omits a lot. There is no account of the different spatial geometries and the big bang is the explosion of a nugget of matter into a pre-existing space. That is not what is portrayed by relativistic cosmologies.
Which of these three cosmologies is ours? The questions is not without some interest. If it is the first cosmology I, then we live in a finite space. If we point in in any direction, after some finite distance, we are pointing at the back of our own heads! Further, just as the universe has a finite past bounded by the big bang, so there is also an end in our future. The entire universe will collapse down onto itself in a "big crunch". In cosmologies II and III neither of these results obtain. Cosmology II, however, is the only one in which the geometry of space on cosmic scales is Euclidean.
The value of the critical density is extremely small: 10-29grams per cubic centimeter of space. That is 0.00000000000000000000000000001 grams per cubic centimeter. That is very little indeed! It corresponds to roughly 5 hydrogen atoms only in a cubic meter of space. That sort of vacuum is extremely hard to achieve with laboratory equipment on earth.
|Here's another measure of how small it is. Take
one fifth of a teaspoon of water, which is
roughly 20 drops. (That amounts to one gram.) How widely spread must it
be in order to match the critical density? Guess! What if we take those
20 drops and spread them over the volume of the Astrodome? Not even
close. Think bigger. What about 20 drops spread over the volume of
earth? Better, but still too small. That density is still 100 times too
big. Those 20 drops of water need to spread over one hundred earth
volumes if their dilution is to match the critical density!
That, at least, is what my sums show. The radius of the earth is about 6,366,000 meters. So its volume is 1.08 x 1021 m3, which comes to 1.08 x 1027 cubic centimeters. So one gram spread over this volume is still roughly 100 times too dense.
Since this critical density is so small, you might think that our universe must have an average density more than critical. That would be jumping to conclusions. What counts is the density of matter averaged over all space. So we need to take the matter of earth and spread it over the vast emptiness of space between stars and galaxies. And then the calculation gets more complicated because of the steady accumulation of evidence that a very substantial portion of the energy of our universe is "dark," so its existence is actually inferred indirectly from the gravitational effects it produces.
The upshot is that the average density of matter comes out very close to the critical. Indeed the astonishing and maddening result is that the more accurately it is measured, the closer our density gets to the critical value. So we remain unable to say which of the cosmic scenarios above is our own.
The suspicion is growing that our density may be exactly the critical density. It seems too much of a coincidence that of all values our matter density could have, it just turns out to be so close to the critical density. So the supposition is that there might be some cosmic process that has driven the matter density to this value. So called "inflationary" cosmologies posit an early phase of very rapid cosmic expansion that would have the effect of driving the matter density towards the critical.
Copyright John D. Norton. March 2001; January 2007, February 16, 23, October 16, November 10, 2008, March 31, 2010; January 1, 2013; March 18, 2015. January 3, 2016.