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Ooops - fixed.I think that should read ##w=\tanh^{-1}\beta=\tanh^{-1} v/c##

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- #26

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Ooops - fixed.I think that should read ##w=\tanh^{-1}\beta=\tanh^{-1} v/c##

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- #28

bob012345

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Thanks, I own that. It's hard to misplace because it's so large it distorts the spacetime around it.....

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it's so large it distorts the spacetime around it.....

- #30

PAllen

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I heard that MTW disproves the principle of equivalence - it falls faster than everything else.

- #31

bob012345

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Newton's Second Law is relativistically modified as F= gamma^3 ma where m is rest mass. This can be derived from the relativistic form of momentum. A few proper substitutions and integrate over x equals mc^2( gamma2- gamma1). All is well.This brings up a related question. I tried applying the work energy theorem in the relativistic regime. I have no problems with the relativistic kinetic energy difference but the integration force over distance got unwieldy. Then I realized than no books I have seen have relativistic kinematics covered. I don't mean Lorentz transformations, I mean in one frame where the acceleration changes for a fixed force because the mass increases. I couldn't match the integrated work with the change in relativistic energy.

- #32

PAllen

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That form is only if force is colinear with velocity. I assume you know this, but it is a crucial caveat to this formula - which is NOT a general form of relativistic force law.Newton's Second Law is relativistically modified as F= gamma^3 ma where m is rest mass. This can be derived from the relativistic form of momentum. A few proper substitutions and integrate over x equals mc^2( gamma2- gamma1). All is well.

- #33

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Sounds like you probably don't need this, but for those that do not have the book there is a concise but well paced derivation of the relativistic rocket equations in this link.Thanks all. I'm digging into these issues. I've picked up Rindler's book and other sources. Started working through some problems. Thanks again!

- #34

bob012345

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Thanks, yes. I know the vector form but was primarily interested in co linear motion. Interesting that there is parallel inertia and perpendicular inertia.That form is only if force is colinear with velocity. I assume you know this, but it is a crucial caveat to this formula - which is NOT a general form of relativistic force law.

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bob012345

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Try applying conservation of momentum to your question, if the light is increasing its momentum then something else would have to increase its momentum in the opposite direction to keep the total constant. What would that be and how?

- #38

bob012345

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I know, that would be both the ship and the medium the light is slowed in if light could actually be slowed. The gaseous medium would carry the momentum away that the ship gains as the light passes out of the medium and returns to normal. The question is if light can actually be slowed in a way that increases it's momentum greatly? I've read where light can be slowed to meters per second in certain vapors but don't know about the momentum of such light. I've heard that light actually does have a bigger kick inside glass but don't have a good reference.Try applying conservation of momentum to your question, if the light is increasing its momentum then something else would have to increase its momentum in the opposite direction to keep the total constant. What would that be and how?

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- #39

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Imagine creating a region around a ship where c=0.00000001 m/s, or slower if needed, so that we will say that light becomes trapped in the region.

How do light traps behave? For example a mirror lined box with a door which is closed after some light has entered.

- #40

bob012345

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I don't know about 'light traps'. Please enlighten me. Thanks.Imagine creating a region around a ship where c=0.00000001 m/s, or slower if needed, so that we will say that light becomes trapped in the region.

How do light traps behave? For example a mirror lined box with a door which is closed after some light has entered.

- #41

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Two examples of "light traps":I don't know about 'light traps'. Please enlighten me. Thanks.

Trap 1: Vertical light beam enters a box, then the light beam bounces inside the box horizontally. First the light had vertical momentum, then it has no vertical momentum, vertical momentum must be conserved, so the box must now have the vertical momentum that the light had.

Trap 2: An atom absorbs a photon, momentum of the photon becomes momentum of the atom.

Some mathematics:

Energy-momentum relation:

E

So massless energy E has momentum: E/c

And energy that has some mass has momentum: Less than E/c

So E/c is the maximum momentum of energy E.

So massless constant energy can not gain more momentum by changing its propagation speed. It can lose momentum by gaining mass and slowing down its propagation speed.

For example: A system consisting of two parallel light beams collides with a lens, which collision causes the beams to become less parallel. After the collision the system has a rest mass that is moving at some speed lower than c, and the system has less momentum than originally, and the lens has absorbed some momentum.

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- #42

bob012345

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Thanks. The question was what happens when c is smaller, like 1 m/s like in a BEC that envelops the ship. Experimental results are hard to come by and theoretical results are contradictory. There has been a over a century of controversy concerning the momentum of a photon in a medium. I just read a paper that claimed to prove the momentum is the same no matter what the speed of light in any medium. Any thoughts here?Two examples of "light traps":

Trap 1: Vertical light beam enters a box, then the light beam bounces inside the box horizontally. First the light had vertical momentum, then it has no vertical momentum, vertical momentum must be conserved, so the box must now have the vertical momentum that the light had.

Trap 2: An atom absorbs a photon, momentum of the photon becomes momentum of the atom.

Some mathematics:

Energy-momentum relation:

E^{2}=p^{2}c^{2}+m^{2}c^{4}

So massless energy E has momentum: E/c

And energy that has some mass has momentum: Less than E/c

So E/c is the maximum momentum of energy E.

So massless constant energy can not gain more momentum by changing its propagation speed. It can lose momentum by gaining mass and slowing down its propagation speed.

For example: A system consisting of two parallel light beams collides with a lens, which collision causes the beams to become less parallel. After the collision the system has a rest mass that is moving at some speed lower than c, and the system has less momentum than originally, and the lens has absorbed some momentum.

P.S. Photon recycling allows one to use the momentum of a photon over and over as long as there is something close to bounce it back. It's been done in the lab with thousands of bounces, moving kg scale objects with typical lab lasers.

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