HPS 0410 | Einstein for Everyone |

Back to main course page

John
D. Norton

Department of History and Philosophy of Science

University of Pittsburgh

- Friedmann-Lemaitre-Robertson-Walker Spacetimes
- 1. Space is homogeneous and isotropic
- 2. Space is uniformly filled with matter.
- Cosmic Dynamics Follows
- Cosmic Time and the Cosmic Rest Frame
- What the Big Bang really is
- Cosmological Red Shift
- Cosmic Dynamics: Three Possibilities
- A Newtonian Analogy for the Dynamics
- Friedmann? Lemaitre? Robertson? Walker?
- And Ours Is...
- What you should know

*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-Lemaitre-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 is sometimes derived from the "cosmological
principle." In a standard form, that principle asserts that "... broadly speaking ... the universe presents the same aspect from every point except for local irregularities." H. Bondi, Cosmology.
Cambridge University Press, p. 11It strikes me as a little dangerous to label a vaguely formulated, happenstance regularity as a "principle." We generally reserve that term for laws of nature or perhaps just for conditions that are not lightly challenged. Whether the universe is roughly homogeneous and isotropic is something that should be determined by observation. It should not be legislated in advance. We saw in the last chapter that those observations seem always to need to move to larger scales before we can secure homogenity and isotropy. |
The distinguishing of this loose idea of sameness
as a principle derives from the cosmological work
on E. A. Milne in the 1930s. He developed a cosmology in
which spatiotemporal relations were induced from considerations of
light signaling, somewhat like Einstein's analysis of special
relativity of 1905 and using analogous principles. His "cosmological
principle" was modeled on Einstein's principle of relativity and
read: "...if A and B are two equivalent particles of
the system, then the description of the system by A (in
terms of his clock-measures or associated coordinates) coincides
with the description of the system by B in terms of B's
clock-measures or associated coordinates."E. A. Milne, Relativity,
Gravitation and World-Structure. Oxford: Clarendon
Press,1935, p. 60. |

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:

Spherical Positive curvature |
Flat, Euclidean Zero Curvature |
Hyperbolic Negative Curvature |

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.

At any cosmic moment, these particles of dust
fill space. Over time they trace out a bundle of
non-intersection, geodesic curves that fill the spacetime. Once
again, this idea has been elevated with the title of "postulate" or
"principle" as in "Weyl's Postulate." It
is named after the mathematician and relativist Hermann Weyl, who
used the assumption in his work. Once again there is a danger in assigning the term "postulate" or "principle." It makes the assumption look like a fundamental law of nature, whereas it is clearly only an accident of our particular universe. Worse it is one that we have seen violated here and there through the observation of colliding galaxies. Here's an image from the NASA website of spiral galaxies NGC 2207 and IC 2163 colliding. |
The notion that there is a principle here seems to
have grown organically. It was implicit in much cosmological
theorizing in the 1920s and 1930s. R. C. Tolman, in his influential
1934 text, Relativity, Thermodynamics and Cosmology
(Oxford University Press), writes of "An alternative hypothesis
suggested by Weyl and ... by Robertson... the nebulae [galaxies] in
the actual universe are to be regarded as lying on a coherent pencil
of geodesics which diverge from a common point in the past." (pp.
356-57).A quick scan found one of the earliest uses of the term in a textbook only in the 1970s: WEYL's postulate: The particles of the substratu (representing the nebulae) lie, in the space-time of the cosmos, on a bundle of geodesics diverging from a point in the (finite or infinite) past. R. K Pathria, Theory of Relativity. Pergamon, 1974, p.
266. |

.

The second type of matter is radiant matter. It is the type of matter Penzias and Wilson found to comprise the cosmic microwave 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 ? 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, decreases.

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.

The term
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 defined. For

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-Lemaitre-Robertson-Walker spacetimes give us three possibilities, cataloged below as I, II or III. (Recall that we have the special case of ? 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^{-29}grams
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.

Cosmology |
I |
II |
III |
---|---|---|---|

Average mass density | Greater than critical | Critical | Less than critical |

Geometry of space | Spherical positive curvature |
Flat, Euclidean zero curvature |
Hyperbolic negative curvature |

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:

Scale factor R |
very roughly equals |
Hubble age of universe |
x | speed of light |

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.

http://publicdomainvectors.org/en/free-clipart/Explosion-vector-illustration/7296.html

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.

These are the names attached in our history to the spacetimes of big bang cosmology. Four names is a lot. How is that four theorists discover the same thing at the same time? The explanation is simply that its time had come and the result was there for anyone with the theoretical competence to pick it up.

It was an exciting time in the
history of cosmology. In the years around 1930 when these
theorists made their discoveries, two ideas came together. __ First__
was Einstein's general theory of relativity. Since Einstein's first
universe model of 1917, it was clear that general relativity had opened
new theoretical possibilities for theorizing on the universe as a whole.

That way of telling the story oversimplifies the situation. It was a time of energetic theorizing and striking creativity. Everyone had their own program, their favored avenues and approaches and their pet aversions. The computation of the "the FLRW spacetime" arose naturally as a small puzzle to be solved within this chaos of different ideas and approaches.

One indicator of the distance between them and us is the treatment
of the initial "big bang" singularity. We take the singularity as an
automatic element in the FLRW spacetimes. It has long since ceased
to trouble us. This was not the attitude at that time. The idea that there was a singularity in spacetime in our quite recent past was bothersome and worse. Arthur Eddington, an influential figure of the time in astronomy and cosmology, was most colorful in his hesitations. He wrote: “Philosophically, the notion of a beginning of the present order of Nature is repugnant to me.” Arthur Eddington “The End of the World (From the Standpoint of Mathematical Physics)” The
Mathematical Gazette, 15 (1931), pp.
316-324 on p. 319. [possibly] reprinted as Nature, vol.
127 (1931) p. 447–453. |

Georges Lemaitre was bolder. But not at first. In 1927, he speculated about a universe that melded Einstein's static universe of 1917 and the expanding universe of de Sitter. In Lemaitre's model, as we go back arbitrarily far into the past, the universe came arbitrarily close to an Einstein universe. It was for most of its past history an Einstein universe, near enough. However the Einstein universe is unstable. Being near enough to an Einstein universe cannot persist. The deviation from it, no matter how small, must grow. And grow it did in Lemaitre's model, until the universe had transitioned to a de Sitter spacetime, near enough.

By 1931, Lemaitre's theorizing had become bolder.
In a letter to the journal Nature in 1931 (vol
127, p. 706), he directly addressed Eddington's suggestion
that a beginning is repugnant. In response, he
speculated--wildly--of a cosmic beginning consisting of just a
single quantum, in the sense of quantum theory. With this initial
state, he surmised:"the notions of space and time would altogether fail to have any meaning at all." He concluded with a repudiation of Eddington's sense of repugnance: "If this suggestion is correct, the beginning of the world happened a little before the beginning of space and time. I think that such a beginning of the world is far enough from the present order of Nature to be not at all repugnant." |

The wrangling over the acceptability of an initial cosmic singularity persisted. It is even enshrined in the name. Fred Hoyle was one of the three founders of the steady state cosmology, in which the universe expanded constantly, but still with an infinite past. On March 28, 1947, he spoke on BBC Radio's "Third Program," where he described the initial singularity of the competing view as supposing the origin of the universe in a "big bang." Hoyle denied his intent was to disparage the notion, but his targets took the term as a badge of honor and popularized it. |

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^{-29}grams 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 10 ^{21} m^{3}, which comes to 1.08 x 10^{27}
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.

- How Friedmann-Lemaitre-Robertson-Walker spacetimes form the basis of modern big bang cosmology.
- How they are picked out of all possible spacetimes by a few conditions on space and the matter is contains.
- What is cosmic time and the cosmic rest frame.
- The three types of universes in FLRW cosmology and what decides between them.
- The Newtonian analogs for big bang cosmology.
- That the big bang is not a moment in time.

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. January 18, 2017.