BurtJordaan » 31 May 2017, 10:37 wrote:In the next part we will use the equivalence between gravity and acceleration to show that this very same graph can also be used for gravity, and that the idea of a black hole then acquires some reality. ;)
Here is a side by side comparison between Bob's inertial frame and Alice's accelerating frame, where Alice experiences a 1g proper acceleration to the right.
- Comparison between Bob's inertial frame and Alice's accelerating frame.
In the accelerating frame (right), Alice's space-propertime (SPT) path has been 'straightened' and in the process her reference frame had to be distorted so that all the horizontal grid lines converge at -1 lyr. This is just the inverse of the 1g acceleration (if it was 2g, the converging would have been at 1/2=0.5 lyr).
This also means that Bob's SPT path had to be "compressed", both in distance and time, as seem by Alice's accelerating frame. It can also be viewed as if, due to her acceleration, Alice "sees" the original rest coordinates as tilted into a 3rd dimension (into the screen), so that Bob's SPT path is still straight and of the same length, but has moved around something resembling part of a cone and is then projected back onto the screen. More about that below.
The most interesting aspect is that Bob's clock would keep on reading his normal time and can be read off the chart along the sloped grid lines. Let us for a moment imagine that Alice's spacecraft was 0.33 lyr long, with Alice in the front and her colleague Tom riding in the back. Bob would then have passed Tom when Alice reached 1yr on her clock. Tom could have peeped into Bob's cockpit and have read his clock, which would have said 1.18 yr. However, Tom's own clock will read 0.67 yr at that event.
Now we have three clocks that all started with 0, but giving 3 different reading for essentially the same event. Clocks running at different rates? No, not quite - it is rather a case of clocks running at the same universal rate, but have covered different SPT paths, just like we had before.
This is easily comprehended if we consider an additional spacetime dimension for the right-hand diagram, directly into the screen, so that the SPT paths can be tilted (or curved) into that 3rd dimension. The SPT paths of all three (Alice, Bob and Tom) are of equal length in this 3-D spacetime. These paths are tilted into the screen at angles depending on their distances from the (coordinate) singularity at x = -1.0. Even Alice's path is already tilted relative to observers to the right (in front) of her, but more about that later.
Remember, all this is just how Alice is observing things in her accelerated frame of reference. She essentially observes curved spacetime all around her. I will try to find a simple 3-D package that can take my data points and present them in a 3-D view from different perspectives. Anyone knows of a simple, preferably free-on-web utility that can do this? It will be cool way for comprehension of what's going on.
Now for the (slow) transition to gravity. The scenario sketched above is exactly equivalent to Alice sitting at a constant distance from a mass concentration that causes a
uniform gravitational field^{[a]} in her vicinity, trying to accelerate her towards the singularity at 1g. Alice has a rocket to keep her stationary in this position in the field, and she feels the same continuous 1g acceleration as before. The inertial Bob will simply be free-falling towards the singularity and feel no g-forces, as before. Tom, in the back of the rocket will actually feel an acceleration of 1.5g, because he is only 0.67 lyr from the singularity. He would have felt the same 1.5g acceleration in the non-gravity situation. Everything is simply equivalent.
With gravity having entered, the singularity now becomes real, well sort-of... It is not yet Einstein's gravity (we will get to that in the next part), but unless Bob has concealed a reserve rocket somewhere, he will eventually (in his own frame) reach the singularity, but first being stretched ("spagettified") and then crushed. What then happen to the atoms and particles of his body, we do not quite know...
In the next part we will see that in the 1912-14 period, Einstein was essentially still viewing gravity like the principles stated above, but he used the math, not SPT diagrams, AFAIK. He still did not consider spatial curvature in his equations, which came later. He finished the first version of his full theory of general relativity (GR) in Sep. 1915 (published Nov 1915).
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Note [a]: Uniform gravity is a first order approximation to full relativistic gravity. It simply means that higher order terms in the full equations are ignored. It works perfectly fine when the gravitational field is not too intense, like in most places in the solar system.