HPS 0410 | Einstein for Everyone |

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John D. Norton

Department of History and Philosophy of Science

University of Pittsburgh

The arrival of Einstein's general theory of relativity marked a rebirth of interest and work in cosmology, the study of the universe on the largest scale. Within Newtonian theory, cosmology had reduced essentially to one question: just how is matter distributed within an infinite, Euclidean space. Einstein's theory made the question much more interesting. For now cosmologists had to contend with many possible matter distributions, many possible geometries and many possible dynamics for both.

What connected all these together was Einstein's gravitational field equations. They specified just which matter distributions could go with just which geometries and how the whole system might evolve over time. One way to think of these equations is as a law that our universe must satisfy. Another is to imagine them as a selection rule. Among all conceivable universes, only some will satisfy Einstein's equations. These universes are the ones that we designate as possible universes, where "possible" now just means "licensed by Einstein's theory."

I like to think of these possible universes as each comprising a page of a great book. Metaphorically, Einstein's gravitational field equations are that book. We shall now turn to reading that book. Just what sorts of universes are possible according to Einstein's theory? As we flip from page to page we will see some quite interested universes. Among them we hope to find out own. |

Minkowski spacetime is the spacetime in which special
relativity holds. This is the simplest all solutions of Einstein's equations.
It arises in when the *unsummed* curvature of
spacetime is zero. It is the case of no gravity. If the
*unsummed* curvature is zero, then the summed curvature must also be
zero. Einstein's equations will be satisfied if the universe is free of
mass-energy. For this reason, Minkowski spacetime is an unrealistic candidate
for our universe. We know our universe has matter in it!

That does not make the structure of a Minkowski spacetime uninteresting. The situation is similar to one we encounter on our earth's surface. We know that this surface is a sphere in the large and that a non-Euclidean geometry must be used to analyze it. Yet in any small patch--such as an area of land the size of a city--we can ignore the curvature and apply Euclidean geometry without appreciable error. Correspondingly, for many applications, Minkowski spacetime is a sufficiently good approximation in the small.

It will fail, however, whenever, gravitation is strong; that is, when the curvature is large. In the vicinity of the sun, gravitation is not strong. So there we manage to extend the use of Minkowski spacetime by pretending that curvature effects of gravitation are really due to a new field, the gravitational field.

The more interesting spacetimes come from other pages of Einstein's book. We find them by "solving" Einstein's gravitational field equations. That just means that we find spacetimes whose summed curvature matches the matter density in the precise way that Einstein's field equations demand. Minkowski spacetime is the simplest solution. Finding others is a formidable mathematical challenge and only the simplest of solutions are easy to find. So when a new solution is found, the new solution is generally named after whoever found it. Finding a solution is really discovering a new, possible universe. So the discoverer's name is then attached to that universe: an Einstein universe, a Reissner-Nordstroem spacetime, and so on.

While the activity of solving Einstein's equations is very hard, the process in conceptual form is quite easy to describe. The Einstein equations specify how spacetime can be locally, that is, at any one point. They say this much curvature always goes with this much matter. To solve the Einstein equations is merely finding a way of distributing curvature and matter over spacetime so neighboring points mesh correctly.

A rather good analogy is to the solving of a jigsaw puzzle. The Einstein equations give us an endless supply of small pieces. They are how spacetime can be in infinitesimally small patches. Each piece has the right combination of curvature and matter. |

Solving the Einstein equations corresponds to
finding a combination of pieces from the supply that can be fitted together. |

The simplest possible solutions are "homogeneous"--that means that they are everywhere the same. In the jigsaw puzzle analogy, that means that the spacetime has to be put together from repetitions of the same piece over and over and over. In the case of a Minkowski spacetime, that piece is just one that has no curvature and no matter. |

This was one of the first interesting, exact solutions given for Einstein's equations and was computed by the German astronomer Karl Schwarzschild. (He died shortly afterwards in World War I at the front.) It is of a universe which looks like a Minkowski spacetime as you get close to infinity in space. It has a central point in space around which all the curvature is distributed symmetrically. It is unchanging in time.

This spacetime is taken to be a good approximation for the spacetime around the sun (as long as we neglect that the sun rotates and has some electric charge).

In the jigsaw puzzle analogy, the solution is put together from pieces that are flat like those of the Minkowski spacetime in regions that are far from the central point. The pieces get more curved as we approach the central point, but their curvature always respects the rotational symmetry of the space around the sun.

The Schwarzschild spacetime is a good approximation of the spacetime around our sun. But does it work for the whole universe? It would work if all the matter of the universe were located in just one island in an otherwise empty space. However, when Einstein started to contemplate these possibilities shortly after completing his general theory of relativity in 1916, he did not like this possibility for two reasons.

First, the astronomical information available to Einstein at that time indicated that the universe was filled with a roughly uniform distribution of stars. So empirically, the model was wrong.

Second, there was a deeper theoretical worry. General relativity had shown how matter fixes the (summed) curvature of spacetime. Einstein liked that notion a lot. It reduced the arbitrariness of a spacetime. Why did the geometry curve this way here and that way there? It did so because the matter distribution went this way here and that way there. The theory reduced the number of arbitrary stipulations that needed to be made in building a picture of our spacetime.

Einstein liked this idea so much that he elevated the idea to a principle. He demanded that the whole geometry of spacetime must be fixed by the matter distribution. That is stronger than what he had up to then. For the Einstein equations only required the matter distribution to fix the summed curvature at each event in spacetime. Einstein called this stronger requirement "Mach's principle," since it reminded him of epistemological analyses of space and time undertaken by the physicist-philosopher Ernst Mach.

The difficulty is that a Schwarzschild spacetime has a property that is not fixed by the matter distribution. That is its flatness at spatial infinity. There is no matter in realms remote from the center of the spacetime, so there is no matter to determine that flatness. It is something that we have to demand in addition.

In the jigsaw puzzle analogy, the problem is this: near the center, we know that we need to lay down pieces of spacetime that respect the rotational symmetry of spacetime around the central mass. But when we get far away from that central island, what sorts of piece are we supposed to lay down? In a Schwarzschild spacetime, we lay down pieces that look more and more like those found in a Minkowski spacetime. But that is now a choice we are making. Nothing about the matter in the central island forced us to make it. That arbitrariness is what worried Einstein.

If you have been reading
attentively, you will have noticed that I have been careful to say only that
this arbitrariness worried *Einstein*. It is not clear that *we*
should we worried about this arbitrariness. Historically, Einstein's
imagination was grasped by the idea of the matter distribution fixing
spacetime geometry completely. It would be a pretty outcome and one that
Einstein found very helpful in guiding his construction of new theories. If
those theories survive empirical testing, we might even discover that his
idea happens to be true. Somewhere, somehow Einstein came to believe not just
that it might happen to be true. He came to believe that it has to be true. This is a much stronger claim. In
reptrospect, it hard to see how to justify it. What if the matter of the
universe were all collected in one big central island. What is wrong with
spacetime remote from the island being flat? The universe has contingent
features. Why can't that be one of them?

How could one avoid the need to stipulate the properties of spacetime at infinity? In 1917, Einstein came up with an ingenious escape: obliterate spatial infinity! By adding an extra term to his gravitational field equations, Einstein found a simple solution of his augmented field equations. ("Augmented"? What is that about? It refers to the famous λ. See below for an explanation.) It contains a uniform matter distribution that approximates a uniform distribution of stars. That matter is at rest and the geometry of a spatial slice is unchanging with time.

Space, however, curves back onto itself so that it is
spherical. That is, space has the geometry of
5^{NONE} with positive curvature. In such a space, there is no
infinity at which to stipulate the properties of space and time.

If one pictures just one dimension of space, then the universe looks like a cylinder. Spacetime resides just in the surface of the cylinder. The vertical lines are the world lines of the stars at rest. The one spatial dimension is wrapped back onto itself; the time dimension is not. Each spatial slice at a particular time appears as a circle; if we could represent all three dimensions of space, we would somehow have to replace the circle by a complete sphere of three dimensional space.

The Einstein universe is an especially simple universe. It is homogeneous. That means that, like Minkowski spacetime, it is geometrically the same at every event. It is also spatially isortropic, which means that it is the same in every spatial direction. In the jigsaw puzzle analogy this homogeneity means that the spacetime is assembled from just one sort of piece, used repeated to build the entire spacetime.

Something very important passed by very quickly just now. The Einstein universe turns out not to solve Einstein's gravitational field equations of 1915. In order to accommodate the new cosmology, Einstein had to make what appeared to be a somewhat arbitrary adjustment to his gravitational field equations.

In their original form, they said

summed curvature
of spacetime |
= | matter density |

We saw in the chapter on general relativity what these equations require inside a uniform matter distribution. That was our first illustration--a mass falling in an evacuated tube inside the matter distribution of the earth. There we saw that there is a positive curvature in the space-time sheets of space time.

The situation is the same with objects within the uniform matter distribution of the Einstein universe. Einstein's original gravitational field equations call for positive curvature in the space-time sheets. That means that his field equations are calling for the same sort of dynamics as we saw for masses falling freely in a tube drilled through the center of the earth. All the matter of the universe should be accelerating towards the other neighboring pieces of matter, just as the neighboring masses in the tube accelerate towards each other. That is, all the matter of this universe should be undergoing everywhere an inward gravitational collapse, perhaps delayed only by an initial outward velocity.

The trouble is that there is no curvature in those sheets in an Einstein universe; the spatial slices remain unchanged through time. There is no convergence or divergence of the points of the matter distribution.

Einstein's resolution was to modify his field equations in a way that would no longer call for this particular curvature. That is, he put another term into the equations that supplied the missing curvature. The real justification was essentially only that it gave him the result he wanted, the admissibility of his new universe. The term added was Einstein's celebrated "cosmological term" or just "lambda" λ. It is a constant term added to the equations, which means that it is the same at every event.

His gravitational field equations now read:

summed curvature
of spacetime |
+ | λ | = | matter density |

As before, each of these
quantities is really a 4x4 table of numbers. The λ table is constant
in the sense that it is the same at every event in spacetime.

At the time, it seemed like a good idea. But Einstein very soon came to regret the addition, which he saw as harming
the formal beauty and simplicity of the equations.

Einstein also almost immediately became embroiled in a dispute with the Dutch astronomer, de Sitter. Einstein had hoped that augmenting his gravitational field equations with the cosmological term would preclude empty universes without matter. De Sitter showed that the augmented equations admitted a cosmology with no matter density, contrary to Einstein's expectations. It was an odd spacetime--now called "de Sitter spacetime"--that is everywhere expanding although there is no matter in it. That means that any two tiny test masses that somehow found their way into the universe would accelerate away from each other, whereever they were located.

De Sitter's spacetime may seem an elaborate
construction. It turns out, however, to be the simplest spacetime after the flat Minkowski
spacetime. It has constant curvature--that is, it has the same
curvature at every event. To see how simple it is, recall our original recipe for generating curved spaces. The simplest case was a flat Euclidean surface. We then generated a two dimensional spherical space by looking at the surface of a three dimensional sphere in a three dimensional space; and we generated a three dimensional spherical space by looking at the surface of a four dimensional sphere embedded in a fourth dimensional space. This procedure in the context of a Minkowski spacetime gives us a de Sitter spacetime. We take a five dimensional MInkowski spacetime (one time dimension, four spatial dimensions). In it, we construct the analog of a sphere; it is the four dimensional hyperboloid shown in the figure. In a Minkowski spacetime, it is a surface of constant curvature, the analog of a sphere in Euclidean space. The four dimensional surface of that hyperboloid is the de Sitter spacetime. One can then easily see how the de Sitter spacetime solves Einstein's gravitational field equations augmented with the cosmological term λ. Since de Sitter spacetime has constant curvature, its summed curvature is everywhere the same. So we generate a solution of Einstein's augmented equation merely by picking that de Sitter spacetime whose summed curvature equals the negative of λ. Then the left hand side of Einstein's equation is zero; the right hand side is also zero since we assume the spacetime to be matter free. |

In retrospect, the extra term Einstein added to his equations had a simple interpretation. A uniform mass distribution, if left to itself, ought to collapse under gravitational self-attraction. That is the physical interpretation of the curvature of the space-time sheets that the equations of 1915 were calling for. In adding the cosmological term, Einstein was, in effect, adding a cosmic force of repulsion that would cancel out this natural gravitational self-attraction. That way the matter distribution could remain static.

When de Sitter forms a universe without matter, no gravitational self-attraction of matter opposes λ's powers of repulsion. We can still insert minute test masses into this otherwise empty universe to plumb its properties. With λ's repulsive powers only in effect, we find a universe in which test masses flee everywhere from each other.

In 1917, when Einstein proposed his universe, the natural supposition was that the matter of the universe is static on the largest scale. In the 1920s and 1930s, it became clear that this was not so. In fact the matter of the universe is everywhere expanding rapidly and that expansion was adequate to counter temporarily the gravitational self-attraction called for by Einstein's theory. (A good analogy is a stone tossed into the air. Its initial, upward velocity overcomes the downward pull of gravity, but only temporarily.) When these dynamic cosmologies emerged, Einstein renounced the cosmological term.

Einstein's renunciation of the cosmological term has not proven to be fatal to the idea, however. It gives cosmologists, eager to match their models to the latest astronomical data, an extra parameter to adjust, so that they can get a fit of their model to new, recalcitrant data.

In that context, there is a popular re-interpretation of the cosmological term. To see it, take Einstein's augmented gravitational field equations in the case in which there is no ordinary matter, so the term "matter density" is 0.

summed curvature
of spacetime |
+ | λ | = | 0 |

Now just take the λ term and move it to the other side of the equal sign:

summed curvature
of spacetime |
= | -λ |

So, where Einstein's original equations used to say "matter density," they now say "-λ." What that means is that Einstein's λ is behaving like an extra sort of matter distributed through space, according to the original equations.

Since we know that λ corresponds to a force of repulsion between matter, it behaves like an odd sort of matter that accelerates the expansion of matter in space. What is odd about it is that all ordinary matter generates attractive gravitational forces. That was the fundamental idea of Newton's original notion of "universal gravitation." As noted above, this now gives some understanding of why the matter-free de Sitter universe is expanding. It is being driven by the repulsions inherent in the cosmological term.

The cosmological constant λ has proven especially useful to recent work in cosmology, for the observed motions of distant matter incorporate accelerated recessions greater than Einstein's original equations allowed. Einstein would not be pleased.

Before we turn to pursue the spacetime that best resembles our own in the next chapter, it is interesting to review how Einstein's theory allows us describe universes in which time travel is possible.

The easiest type of time travel universe looks like a trick that is stipulated into existence. However there is nothing illicit about it. And its great simplicity enables us to refine our intuitions about just how time travel can arise.

Einstein showed us through the Einstein universe that we can curve space back onto itself and thus produce a closed space.

That construction proved a little complicated for Einstein since there are three dimensions of space that need to be accommodated. If we want to do it in the time direction, it is much easier. There is only one dimension of time. The simplest case arises if we wrap up the time direction of a Minkowski spacetime. As before, if we consider only one dimension of space, we recover a cylinder. The spacetime is on the surface of the cylinder. For the new case the cylinder is wrapped up in the timelike direction.

You might wonder if a trick like this is really allowed by Einstein's gravitational field equations. It is. Recall that Einstein's gravitational field equations merely fix how each little patch of spacetime must look. A solution is admissible if each patch connects properly with those next to it. That will happen in this spacetime. In any not too big piece, this cylinder universe is exactly the same as a Minkowski spacetime; each piece connects with the one next to it just as they do in a Minkowski spacetime. That is all that is needed for the spacetime to count as a solution of Einstein's gravitational field equations. |

The timelike curve on the spacetime represents the life of a traveler who stays at one point in space, but passes through time merely by being. Eventually that worldline will wrap all the way around the spacetime and reconnect. At that point, the traveler will meet his or her former self.

The traditional "grandfather" paradox of time travel arises if the latest stage of the traveler (now imagined to be the grandson of the original traveler) were to kill the original one (the grandfather). A contradiction would ensue. With the assassination complete, there would no traveler to pass through time and commit it. So the assassination happening entails that it doesn't happen.

The possibility of such paradoxes has led some to conclude that time travel universes are logical impossibilities. That is too hasty. There is an obvious loophole in the paradox. If the assassination attempt fails, then there is no contradiction.

So that is what must happen in a time travel universe. The grandson's bullet must miss; or the gun misfire; or the grandfather ducks; or who knows what. For if the assassination attempt didn't fail, there would be no assassin to attempt it.

That resolution is, as far as I know, admissible. Many find it objectionable since there seems to be no reason in the physics itself that forces the failure of the assassination attempt. What if the grandson takes all due care, aims carefully with a new gun, and so on? How can we be so sure that the attempt will fail.

We can. The intuitions that tell us it will not fail are honed in a type of universe that is quite different from a time travel universe. In the ordinary time travel free universes, such as we presume we inhabit, local constraints prevail. If the gun misfired, for example, it was because something in the state of the gun immediately prior to to the assassination attempt intervened. Perhaps the grandson passed through a rain shower and a component of the gun began to rust.

In a time travel universe, in addition to these sorts of local constraints, we have a new type: global constraints. These are extra constraints that all processes must conform to in order that distant future and distant past mesh when they meet. These global constraints do not arise in time travel free universes. They are what assures us that the assassination attempt must fail.

We can get an idea of how they work from the jigsaw puzzle analogy for solving Einstein's equations.

First consider a universe without time travel. We start with a row of pieces that represents space in the present instant. Then we add successive rows that correspond with space in successive future times. The pieces we add are constrained only by the local requirement that each piece mesh with those immediately before and after it in time; and those around it in space. |

Now take the case of a time travel universe. All these constraints apply. But, in addition, as we keep adding the successive rows, we will eventually end up going all the way round the space and then the new and powerful constraint will come into force. The last row we add has to be so perfectly built so that it meshes with the past edge of the first row we put in place. That is a global constraint. It means that in our planning of which pieces to lay down, we had to worry about the local meshing of the pieces; and, in addition, we had to select pieces now so that eventually the final meshing of last and first row would work out. |

Here's a simple example in a
different arena of how these sorts of global constraints can work. It is the
arithmetic puzzle, "99." In the puzzle, you are
to start at zero and may add or substract any number you like between 0 and
10, as many times as you like, provided that the numbers that you are adding
or subtracting are always *even* numbers. Is there some combination of
additions and subtractions that will get you to 99?

Locally, there is no obstacle to getting to 99.
If you could somehow get your sum to 97 or to 95, you could complete the task
by adding 2 or 4. Just looking locally at the numbers around 99 reveals no
problems.

Globally, however, there is a constraint that
necessarily defeats your attempts to arrive at 99. Since you start with zero
and may add or subtract even numbers only, your sum must always be an even
number. So you can never get to 97 or 95 or any other number that is an even
number removed from 99. This global constraint assures your failure to solve
the puzzle.

A simple illustration show just how powerful these global restriction on a spacetime can be. Consider just about the simplest possible time travel universe: a universe empty of all matter excepting just one mass. Now pick some time slice. What configurations of the particle are possible?

In an ordinary time travel free universe, at some initial moment of time, we can have the worldline of the mass with any initial velocity.

If the spacetime is a time travel, cylinder universe, we are strangely restricted in the possibilities for this time slice. We could choose a mass at rest. That corresponds to the case of a single worldline that eventually wraps back onto itself. But if we have the mass initially moving, then we must also stipulate that clone masses be distributed in space at uniform intervals. These will be the repeated returns of the single mass as it travels all the way round spacetime and back to the present.

The global constraint says that if we have a moving mass here and now, we must also have a moving mass there and now; and there and now; and so on. That sort of constraint would be incomprehensible in a universe without time travel. What reason of physics, we would exclaim, requires it--just as we ask, what reason of physics requires the grandson's assassination attempt to fail!

There is something that looks just a little fishy about the way time travel is arrived at in the cylinder universe. It does not seem to arise from the physics of the spacetime. It comes from a stipulation on our part that the future wrap back onto past. Einstein's theory seems only to get involved in so far as it raises no objection. There is nothing wrong with this way of introducing time travel, of course.

It is nice to know, however, that time travel also can arise more naturally. The Goedel universe is one such example. This solution to the Einstein equations was arrived at by the famous logician, Kurt Goedel, in the 1940s, when he was a colleague of Einstein's at the Institute for Advanced Study in Princeton, and published in 1949.

The Goedel universe is a solution of Einstein's equations
with the cosmological term. Its signature property is that it contains closed timelike worldlines. As a result, it is hard to
pick out a single timelike direction globally in the spacetime. Rather, we
can get a feel for its spacetime properties by taking just a single two
dimensional slice of it. It will become clear that this is *not* a
spacelike slice.

If we consider some observer in the middle of this slice, the observer will find all the matter in a great cosmic rotation around them. (For this reason, the Goedel universe cannot be ours. We don't see such rotation.) The reason for the rotation lies with the structure of spacetime itself. As we consider positions in the slice further away from the observer, the light cones start to tip over. So if we consider a large enough chunk of the slice, we can find a timelike curve that loops back onto itself. It forms a closed timelike curve, the hallmark of universes that admit time travel.

The timelike curve is not a geodesic; it represents the trajectory of an accelerating spaceship. To achieve time travel, the spaceship would need to accelerate quite considerably. Most interestingly, the Goedel universe uses no stipulations about past wrapping back onto the future to achieve the possibility of time travel.

There are other universes that admit time travel. Often rotation is involved. Spacetime around an infinitely long, very dense, rapidly rotating tube of matter admits closed timelike curves, for example. Some of the most fascinating of the time travel universes are those in which one part of spacetime is connected to another by a wormhole. That is just a tunnel of spacetime that provides an alternative route from one part of spacetime to another.

- The analogy between solving Einstein's gravitational field equations and the solving of a jigsaw puzzle.
- How Einstein's cosmological constant λ modifies his gravitational field equations.
- The basic characteristics of various solutions of the Einstein equations:
- Minkowski spacetime
- Schwarzschild spacetime
- Einstein universe
- de Sitter universe
- cylinder universe
- Goedel spacetime

- How time travel can arise in relativistic cosmologies.

Copyright John D. Norton. March 2001; January 2007; February 16, October 15, 27, 2008.