Suppose there is a guy in a closed off room floating somewhere in the deep regions of space, far enough away from every object that there is no gravity. He is basically floating in the middle of the room, because there is no gravitational attraction moving him anywhere. If he pulls his wallet out and lets it go, it will just float right next to him. If he gives it a little push, it will move in the direction he pushed it until it hits the wall. This is all possible because the man is in an inertial reference frame- i.e. Newton's laws accurately describe what happens in his room.

Now, let's say outside the room, unbeknownst to the man, there is a rope attached to a giant hook on the 'ceiling.' Then some 'being' starts pulling on the rope with a constant force, and so the room, from anyone watching on the outside, would be constantly accelerating in the direction being pulled.

However, inside the room, the man will suddenly feel himself being pulled down to the 'floor' of the room. If he now takes off his watch and lets it go, instead of just floating like his wallet did before, it will 'fall' straight down to the ground. The man is completely justified, therefore, in assuming that he is now in a gravitational field, and all objects in the room will 'fall' down just like his watch does.

The point is that being uniformly accelerated "feels" exactly like being in a gravitational field. This led Einstein to the famous "equivalence principle," which states that a uniform acceleration and a gravitational field amount to the same thing, more or less.

But why should this mean that space-time is "warped?" How does Einstein get from "uniform acceleration and gravitation are indistinguishable" to "space-time bends?" Hopefully this post will shed some light on that. I can't explain it in any real detail, but I can repeat an example that shows why Euclidean geometry had to be thrown out the window as a geometrical theory of our universe.

Here is the example, a though experiment Einstein devised:

Let's say there is a very large circular disk, rotating at some fast rate. On this disk, very near the center is a small circle drawn. Further, assume there are two observers: one in the middle of the giant disk, and one on the outside. The observer on the outside is in an inertial frame. Here is what the disk looks like:

A little circle painted on the center of the big circle.

Now, imagine the whole thing rotating really fast. If you are clever, you will notice that points on the outside of the circle are moving at a greater speed than points further inside the circle. That is because in order to remain attached to the circle, those outside points have to cover more ground in the same amount of time as points closer to the middle of the circle do.

*(This is pretty obvious if you've ever watched a CD or record spin, or if you've ever been hit in the head by a stick- it hurts a lot more to be hit by the edge of a stick moving in an arching path than to move in close and instead get hit by, say, the handle, as the edge of the stick moves past you.)*
Okay, so now the real test: the two observers will measure the radii and circumference of each circle, and see if both of them give the ration of

π. If they both give this value, than we can safely assume that the laws of Euclidean geometry hold. If either one of them do not, then we can only assume that the laws of Euclidean geometry hold for the frame in which the ration of

π is measured, and we

*cannot* assume that the laws of Euclidean geometry hold for both reference frames.

Alright, so the outside observer measured the length of both radii and both circumferences. He then divided the two, and got, as he predicted, the value

π. Obviously, because he was measuring circles.

**So it can be reasoned that the outside observer's space-time is consistent with Euclidean geometry.**
Now, then the circle starts rotating uniformly, and the man on the circle begins measuring. He measures the radius of the small circle, and then the circumference of the small circle. Remember that the velocity of points inside the small circle are much smaller than the velocity of the points on the edge of the big circle, so his measurement of the radius and circumference are each respectfully the same as the measurements found by the outside observer. The observer on the circle then computes the ratio, and correctly calculates the number

π for the small circle.

Then he begins to measure the radius of the big circle, starting from the inside and working his way out with his measuring rod, one ruler length at a time. Note that there is no velocity in the direction of the radius- the disk is spinning, not zipping along horizontally- all of the velocity is directed in a circle. Because of this, there are no relativistic affects on his ruler in the direction of the ruler's length (the ruler's length remains constant), so he again measures the exact same length for the large radius as the outside observer did.

He then begins to measure the circumference of the large disk.

**Here is where things get interesting.** According to special relativity, objects moving at near the speed of light will have their lengths contracted (shortened)

* in the direction of motion*. Since the man is now measuring along the same direction that the circle is spinning (the circumference), according to special relativity his ruler should be contracted. If he is measuring the same circle with a shorter measuring rod, then he will correctly measure the circle to have a larger circumference than the outside observer did.

But if he measured the SAME distance for the radius, but a DIFFERENT distance for circumference, then how can he get the same ratio that the outside observer did? The answer is that he CAN'T. 4/20 ≠ 4/25. It's basic math. Therefore, the observer on the disk will NOT get

π as the ratio of circumference to radius.

**Therefore we can reasonably conclude that the space-time of the observer on the disk is NOT consistent with Euclidean geometry. **
He would need different geometrical laws to describe space-time in his reference frame.

And in order to describe the entire situation (both observers), we would need geometrical laws that behave like Euclidean geometry in inertial frames, but behave differently in accelerated frames.

And now if we combine this new result with the first part of this post, the equivalence of uniform acceleration and gravitation, we could derive a geometry that behaves like Euclidean geometry in inertial frames (and free fall frames), but behaves differently in both accelerated frames and in gravitational fields.

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