Properties of space and time.

Up to here I've talked little of the implications of the Special Theory of Relativity on the General Theory of Relativity, I have only argued that in special relativity time and space are interconnected. In a separate discussion I argued that gravity alters space. In this section I will use what we know about length contraction together with the equivalence principle to determine how space is altered by gravity and to show that it is this deformation of space that is responsible for the gravitational force.

Imagine two identical disks one of which is made to rotate uniformly as in Fig. 7.14; each disk has its own observer provided with a meter stick, labeled l and l₀ in the figure. The disks are so constructed that when overlapping their circumferences match. The rotating meter stick is continually moving along its direction of motion so that l will be length-contracted with respect to l₀ (Sec. 6.2.4), so a larger number of l will fit in the circumference. This means that the rotatingh observer measures a longer circumference than the non-rotating (inertial) observer.

 

Figure 7.14: A rotating vs a non-rotating disk. The bit labeled l in the rotating disk is shorter, due to length contraction to the corresponding bit l₀ in the non-rotating disk.  

Consider now a radius of the disks. This is a length that is always perpendicular to the velocity of the disk and it is unaffected by the rotation: both disks will continue to have the same radius (see Sect. 6.2.4).

So now we have one non-rotating disk whose circumference is related to the radius by the usual formula, circumference = 2 $\pi$ × radius, and a rotating disk whose observer measures a larger circumference but the same radius. In the rotating obnserver the formula does not hold!

How can this be? Isn't it true that the perimeter always equals 2 $\pi$ × radius? The answer to the last question is yes...provided you draw the circle on a flat sheet of paper. Suppose however that you are constrained to draw circles on a sphere, and that you are forced to measure distances only on the sphere. Then you find that the perimeter measured along the sphere is smaller than 2π × radius (with the radius also measured along the sphere, see Fig. 7.15).

 

Figure 7.15: The distance from the equator to the pole on a sphere is larger than the radius. For being constrained to move on the surface of the sphere this distance is what they would call the radius of their universe, thus for them the circumference is smaller than $2 \pi \times$radius and they can conclude that they live in curved space.  

A similar situation is observed in the rotating disk with a similar solution: the reason the rotating observer fids that the circumference is not equal to 2 $\pi$ times the radius is that this observer is in a curved surface. On a sphere we just saw that 2 $\pi$ × radius > circumference, in a saddle-shaped surface 2 $\pi$ × radius < circumference as in the situation we have been looking at.

We conclude that the uniformly rotating disk behaves as a (piece of a) saddle-shaped due to length contraction. So much for the effects of special relativity.

Now let us go back to the principle of equivalence. One of its consequences is that, by doing experiments in a small region one cannot distinguish between a gravitational force and an accelerated system. So if we attach a small laboratory of length l₀ (at rest) to the small section of the perimeter, experiments done there will not be able to tell whether the lab. is in a rotating disk or experiences a gravitational force (remember that a rotating object is changing its velocity - in direction - and it is therefore accelerating!).

Putting together the above two arguments we get

Gravitation curves space and time.

Conversely curved space and time generate effects which are equivalent to gravitational effects. In order to visualize this imagine a world where all things can only move on the surface of a sphere. Consider two beings labeled A and B as in Fig. 7.16, which are fated live on the surface of this sphere. On a bright morning they both start from the equator moving in a direction perpendicular to it (that is, they don't meander about but follow a line perpendicular to the equator).

 

Figure 7.16: Two beings moving on a sphere are bound to come closer just as they would under the effects of gravity 

As time goes on the two beings will come closer and closer. This effect is similar to the experiment done with two apples falling towards the moon (Fig. 7.4): an observer falling with them will find their distance decreases as time progresses; sentient apples would find that they come closer as time goes on.

So we have two descriptions of the same effect: on the one hand gravitational forces make the apples approach each other; on the other hand the fact that a sphere is curved makes the two beings approach each other; mathematically both effects are, in fact, identical. In view of this the conclusion that gravity curves space might not be so peculiar after all; moreover, in this picture the equivalence principle is very natural: bodies move the way they do due to the way in which space is curved and so the motion is independent of their characteristics , in particular the mass of the body does not affect its motion.

 

Figure 7.17: Just as bugs fated to live on the surface of a sphere might find it peculiar to learn their world is curved, so we might find it hard to realize that our space is also curved.  

Now the big step is to accept that the same thing that happened to the above beings is happening to us all the time. So how come we don't see that the space around us is really curved? The answer is gotten by going back to the beings A and B: they cannot ``look out'' away from the sphere where they live, they have no perception of the perpendicular dimension to this sphere, and so they cannot ``see it from outside'' and realize it is curved. The same thing happens to us, we are inside space, in order to see it curved we would have to imagine our space in a larger space of more dimensions and then we could see that space is curved; Fig. 7.17 gives a cartoon version of this.