- #31
Passionflower
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Right! So think about it, an observer free falling at escape velocity, is rapidly approaching the EH, faster and faster. So yes, the closer he gets to the EH, the more the gravitational field will slow down his clock compared to one at infinity. But then, he travels faster and faster, so due to relativity the clock at infinity is going...right...slower and slower. So we got: clock at infinity going faster and going slower, which one will win? Well look at the graph!kamenjar said:
Well we already know the circumference: it is simply 2pi r and we can also calculate the area. Because spacetime is curved the proper distance between two different r values is not r2-r1. We can calculate any distance, area and volume up to the EH and passed the event horizon we can do the same from a particular r value to the EH, but not from and to r=0 which is the singularity. But we can certainly measure the distance in terms of time it takes for a free falling observer at escape velocity to reach the EH and the singularity. For instance, when an observer of about 2 meters of height measures a tidal effect between his head and feet of about 1g, he is about 0.3 seconds away from reaching the singularity.kamenjar said:
Well you certainly can express it in terms of how much time it takes for something to get there. The formula is surprisingly simple. If we take an example where the event horizon is 1 (so the mass is 0.5 meters, mass in gr is defined as a distance by the way) then we have:kamenjar said:
<br /> -2/3\,{r}^{3/2}<br />
Now that formula is not too hard right? So, say a free falling observer traveling at escape velocity is at a given moment at r=20 how many seconds on his own clock does he still have left before it is all over? Well you can type the formula in your calculator to get the answer, the answer is roughly 59.63 seconds. We can also calculate how long it takes to reach the EH, the formula is not that much harder, it is:
<br /> 2/3\,{r}^{3/2}-2/3<br />
So if we calculate for r=20 we get: 58.96
As you can see for a black hole with a Schwarzschild radius of 1 the traveler has only about 0.7 seconds going from the EH to the singularity. That is not very much right?
Well first thing is that you got to say to yourself you are wrong. The distance between every two r values is always finite except for r=0! For instance a spacestation hovering at r=20 will measure the distance to the event horizon to be around 21.7 meters, a little more than, as you would expect, 19 (20-1). Why the difference? Answer spacetime curvature! We can even calculate the volume of space between r=20 and r=1, it is about 34755 cubic meters, more than one would expect if the spacetime where flat, in flat spacetime we would get: 33506.kamenjar said:
Well we can throw a formula at it. That is the beauty of science! Thanks to Karl Schwarzschild we can indeed throw a bunch of formulas and get results!kamenjar said:
Well actually it is not that hard, the hard part I think is for you to let go of some of those convictions that you have then you can learn. It is like the Chinese proverb about the teacup being full, you first have to empty it to get new tea!kamenjar said:
Most important one at this stage for you I think is to realize that the distance between any two r values is finite, except for r=0. So everything is easily calculable all the way up to the value r=0.