# Discussion 4: General Relativity

Reading Assignment: Arny Section 14.3 (pp 424-428), Cosmos Chpt. 8

# General Relativity

By 1915 Einstein had discovered that there should be a sense of relativity that included frames of reference that are accelerating. Hence he was able to include gravity in his new theory.

## The Equivalence Principle

In special relativity we saw that when two people moved relative with respect to one another at constant velocity they could each consider themselves to be at rest with equal validity. When accelerations are involved the situation takes on another face altogether.

Imagine that Bob and Jill are floating around in space and Bob suddenly fires his engine and produces an acceleration up of about 9.8 m/s/s. Now Bob cannot consider himself to be at rest while Jill accelerates away from. He feels weight but he can see that Jill is floating freely. He should be floating free if is he is not in motion and she should feel weight if she were accelerating. It would certainly seem that there is no relativity here. But Einstein thought that the laws of physics should be the same for all observers no matter if they are feeling forces or not.

In 1907 Einstein came upon a great revelation. Whenever you feel weight (as opposed to weightlessness), you can equaly well attribute it to the effects of either acceleration or gravity. This idea is called the equivalence prinicple: The effects of gravity are exactly equivalent to the effects of acceleration.

Imagine that you are sitting in a closed room with the shades drawn. If your room was magically moved to outerspace and was accelerated at a rate of 9.8 m/s/s you would not notice anything different. If you did physics experiments by dropping balls and so forth you would yeild the same results as you did when the room was on Earth.

Return to Bob and Jill. Bob thinks that he must be accelerating upward. But he could equivalently say that his ship is at rest on the Earth's surface, perched on a cliff say, and that is why he feels weight. The reason he sees Jill in free float would be that she is falling toward the Earth. She is in freefall just like the space shuttle astronauts.

The physics of the two situations are exactly equivalent. So the equivalence principle allows us to state that all motion is relative. This will lead us to some startling discoveries about the nature of space and time!

## Spacetime

The physics of the two situations above may be the same, but they sure don't look the same. Why is that. General Relativity would say that it's because we are not looking at the whole picture. We are looking at the 3-dimensional picture and ignoring the time aspect. But what we have discovered thus far with relativity is that space and time are interconnected and bound up into a 4-dimensional spacetime. We cannot see the full 4-dimensional events only their projections into 3-D. So, we must learn to "look" at the full 4-D spacetime.

Spacetime is a 4-D continuum in which the independent directions of motion are up-down, forward-back, left-right, and through time. Events occur in 3 spatial dimensions and at one point in time. In the whole continuum objects are streched out in time as well. If you were a 4-D being in this continuum you could look through time as easily as you look to your right.

In 3-D we can only see projections of 4-D. And the projections can look very different from different point of view. As an example. Imagine a book. Everyone can come measure it and agree on its dimensions. But it can be viewed in projection into 2-D as very many things.

So the heart of relativity is then that different observers are seeing different projections of 4-D spacetime into 3-D space. "Space is different for different observers. Time is different for different observers. Spacetime is the same for everyone." - Taylor, Wheeler, Freeman.

## Curvature of Spacetime

We shall discover that mass/energy causes spacetime around it to curve, and that gravity is the result of an object's path through curved spacetime. But whatever does it mean for spacetime to be curved?

We must look at lower dimensional analogies to get a feel for what this means since it is not only impossible to imagine 4-D objects, but 4-D space curved into some 5th dimension (a hyperspace).

Consider good old 2-D geometry. ....

So what does motion in a straight line mean? It means taking the shortest path. In a curved space that is going to be a curved line. In relativity that will be a line along which you feel no net forces. A force-free frame travels on a straight line.

So if you are falling toward the Earth you are feeling no net forces as you are in freefall, and thus you are on the shortest path toward the center of the Earth. If you wish to get to the other side of the Earth (say from some high poing way up in orbit) the shortest path would not be through the center of the Earth. Once you pass the center of the Earth you are now feeling a net force back toward the Earth. The shortest path is along the orbit (which also has no net forces acting on it).

## Gravitational Time Dilation

Imagine Jill and Bob in a long ship. Bob is in the front and Jill is in the back. Each has a flashing watch perfectly synchronized. The ship begins accelerating. As the ship accelerates Bob will be carried away from Jill's flashes and so they are stretched out in time, thus he sees her time to run slower than his. Likewise Jill is being carried into Bob's flashes making his time seem to run faster than hers. Since acceleration is equivalent to gravity we can imagine the same situation with the long ship on Earth.

Time Travel