Rod shortening of General Relativity

[Pierre007080]

Hi Yuiop, The maths and terminology being discussed is largely above my head. Passionflower's mention of volume has helped me identify my problem with your "double transformation" Prof Susskind used fluid dynamics to demonstrate how the flow divergence was the mathematical analogy to the gravitational field. The acceleration flow field gave me this image of "space flowing" even though he was quick to mention that space does not actually flow. Using this analogy the speed of "flow" can easily be determined by realting the total work done (GMm/r) to the kinetic energy (1/2mv2). Thus v =√ 2GM/r. This is obviously the speed of a free falling test mass and also the escape velocity at that particular radius. If the free falling reference frame were to be seen as the "instantaneous" inertial frame, the large mass would be approaching the test mass at this speed. If the Lorentz transformations for rod shortening are applied by inserting v =√ 2GM/r, then the gravitational rod shortening factor you gave me earlier √ 1-2GM/r emerges. Is this some sort of circular reference??
Why I mention this "space flowing" instantaneous speed is that if the free falling body had no velocity at infinity, the the "space flow" speed would equal that of the body and only one transformation would apply. If not then of course the second transformation would apply to the difference of speed between "space" and the body. Is this too naive to have intuitive significance?

 

[yuiop]

Hi Yuiop, The maths and terminology being discussed is largely above my head. Passionflower's mention of volume has helped me identify my problem with your "double transformation" Prof Susskind used fluid dynamics to demonstrate how the flow divergence was the mathematical analogy to the gravitational field. The acceleration flow field gave me this image of "space flowing" even though he was quick to mention that space does not actually flow. Using this analogy the speed of "flow" can easily be determined by realting the total work done (GMm/r) to the kinetic energy (1/2mv2). Thus v =√ 2GM/r. This is obviously the speed of a free falling test mass and also the escape velocity at that particular radius. If the free falling reference frame were to be seen as the "instantaneous" inertial frame, the large mass would be approaching the test mass at this speed. If the Lorentz transformations for rod shortening are applied by inserting v =√ 2GM/r, then the gravitational rod shortening factor you gave me earlier √ 1-2GM/r emerges. Is this some sort of circular reference??
Why I mention this "space flowing" instantaneous speed is that if the free falling body had no velocity at infinity, the the "space flow" speed would equal that of the body and only one transformation would apply. If not then of course the second transformation would apply to the difference of speed between "space" and the body. Is this too naive to have intuitive significance?

There is a coordinate system called the "river model" or more formally the Gullstrand-Paineleve coordinates, that is similar to what you are talking about. In an informal description of the river model, "spacetime" is pictured flowing into the black hole and has a velocity equal to the speed of light as it passes the event horizon. In these coordinates everything moves relative to this flowing spacetime and an ingoing photon is moving at 2c as it passes the event horizon and an outgoing photon has a velocity of 0c at the event horizon because it cannot go against the current of the river and cannot escape a bit like trying to walk up a fast escalator. In these informal descriptions, they usually add a disclaimer that nothing is actually flowing because they realize that if something was actually flowing it would be an acknowledgment of an ether. One immediate problem with such a description is that it might lead to the conclusion that the time dilation of a stationary clock in Schwarzschild coordinates is due to the flow of this "substance" past the stationary clock. The stationary clock is effectively moving relative to the flowing river and it turns out 9as you have pointed out) that the time dilation of the stationary clock is equivalent to the time dilation a clock moving relative to an observer in flat SR spacetime with a velocity exactly equal to the falling escape velocity of the river. As you already have pointed out this implies that a clock freefalling with escape velocity is stationary with respect to the river substance and should experience no time dilation, but we know this is not true because we have proven in this forum (jn many different ways) that the time dilation of a falling clock is $tau = t*\sqrt{(1-2m/r)}*\sqrt{(1-v^2)}$ where v is the local velocity measured by a stationary Schwarzschild observer at r. That last equation works whether the clock is moving horizontally or vertically and it does not have to be moving at escape velocity. If the clock does happen to be free falling at escape velocity then the time dilation factor is simply $tau = t*(1-2m/r)$

 

[yuiop]

$$\begin{align*} & 4/3\,\pi \,\sqrt {{r_{{2}}}^{5} \left( r_{{2}}-2\,M \right) }+10/3\, \pi \,M\sqrt {{r_{{2}}}^{3} \left( r_{{2}}-2\,M \right) }+10\,\pi \,{M }^{2}\sqrt {r_{{2}} \left( r_{{2}}-2\,M \right) } \\ & + 20\,\pi \,{M}^{3} \ln \left( 1/2\,{\frac {r_{{2}}}{M}}+1/2\,\sqrt {2\,{\frac {r_{{2}}}{ M}}-4} \right) -4/3\,\pi \,\sqrt {{r_{{1}}}^{5} \left( r_{{1}}-2\,M \right) } \\ & - 10/3\,\pi \,M\sqrt {{r_{{1}}}^{3} \left( r_{{1}}-2\,M \right) }-10\,\pi \,{M}^{2}\sqrt {r_{{1}} \left( r_{{1}}-2\,M \right) } \\ & - 20\,\pi \,{M}^{3}\ln \left( 1/2\,{\frac {r_{{1}}}{M}}+1/2 \,\sqrt {2\,{\frac {r_{{1}}}{M}}-4} \right) \end{align}$$

This is the very integral I mentioned in post#25.

I could not resist shortening that equation a little bit to:

$$\frac{5}{6}\pi \left[3r_s^3\ln\left(r+\sqrt{rr_s - r_s^2}\right) + 3r_s^2\sqrt{r_2^2-r_2r_s} +2r_s\sqrt{r_2^4-r_2^3r_s} + \frac{8}{5}\sqrt{r_2^6 - r_2^5r_s} \, \right]_{r=r1}^{r=r2}$$

where rs-2M.

 

[Pierre007080]

After long consideration of the simplistic radial formulae obtained for the theoretical case proposed at the outset of this thread, it appears that the rod shortening and time dilation at a certain gravitational potential are the same as the rod shortening and time dilation that would be obtained by applying Lorentz transformations to the velocity of a body free falling from infinity along the radial. Is there any significance to this fact?

 

[Pierre007080]

Does the modern view of "vacuum" not present sufficient inertial energy in space itself to conform to the flow concept in the "river model"?

 

[Pierre007080]

Hi Guys,
Your previous insights on this thread really helped me to have a naive but practical idea of what GR is about. The next step in my thought process seems to be the EM force.
I would like to pose a question with regard to the radial shortening of a charged object (say negative) being attracted by another "STATIONARY" charge (say positive). Would it be possible to apply the same formulae as the case with gravity as discussed previously in this thread. I realize that the force is much greater than gravity, but the force formula seems to be related (charge in stead of mass).
Perhaps you could use the theoretical example of a proton attracting an electron from infinity to explain?

 

[Dale]

This really should be in a new thread. The title of this thread is not descriptive of your current question.