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This is a curious problem. It looks basic enough to have been considered before,
but I find nothing related anywhere. Background: The gravitational Doppler shift formula is straightforward when the energy change is proportionately small. Tracking energy E, we take E_received = E0 (1 - g dot h/c^2). Following the equivalence principle, we expect this to apply in general to relative gravity fields, whether "real" or caused by relative motion, or even the *combination of both*. However, if you directly work out the change in relative energy for a photon traversing a frame moving at constant velocity in a g-field (ie, elevator), the result is not as given by the Eq. applied to local values of g and h for the elevator K'. This problem would not appear in Newtonian physics. (Let us know if you've heard of anything like this problem, especially with references.) I think this effect is testable with current equipment. Critique is fine, but it won't help unless you show your work. I think sci.astro is appropriate given the GR crowd, and the problem could have cosmological consequences. The point can be summarized as: First, assume you are in a "rest frame" K which follows the normal rules given above, and that once light is emitted it follows consistent rules. Then, consider a moving frame and track the paths and energy changes of photons emitted from one place to another, combining the effects of both gravity and local relative velocity. Photons emitted from the trailing end P1 of the elevator travel farther along the "rest" frame of the gravity field to catch up, and thus suffer greater gravitational Doppler shift in transit than would normally apply with that elevator height. Photons emitted from the leading end P2 of the elevator travel less rest-frame distance than if the elevator were at rest, and show less GD shift. Since the velocities of the ends are locally the same, the velocity part of Doppler shift (which acts at the moments of emission and reception) cancels out, leaving the travel-based discrepancy. (Putting the math simply: the emission and reception Doppler formulae give inverse values, leaving the proportional change from gravitational effect to be solely that determined by the distance the photon travels in K.) Consider also reflecting a photon from the other end of the elevator: the photon is received at a different potential (in K) than when emitted, but combined velocity effects all cancel out. Therefore, it must show a net energy change, which wouldn't happen in "normal" gravity fields. It perplexes me too, but this is the result of directly working things out. I'll work it out in more detail, but using the simple case of light going parallel to g. If we combine Lorentz contraction (which does locally apply in a g-field) and "catch-up" calculations, we get the following for the values of h as actually moved in K, in terms of proper elevator height L0 and its velocity v, which is locally consistent: h = gamma*L0 (1 + v/c), with v signed negative when sent from a leading end and L0 signed negative when light moves down. Things may get more complicated when v approaches c, but at low v it is clear that the top of the elevator will move very nearly this extra margin before receiving a photon, etc. Hence, when we plug this formula into the GD shift we get E_received = E0 [1 - gamma*g L (1 + v/c)/c^2] Since the g' felt in K' is multiplied by gamma (check with transformations), the actual discrepancy versus relative g' is E_received = E0 [1 - g' L (1 + v/c)/c^2]. This seems like it would violate energy conservation, since the photon's energy change does not correspond to the work doing moving the mass-energy in the local g', but remember that when we move the elevator, the impulse from photon emission and reception must be accounted for. This problem raises questions about the equivalence principle also, since we'd expect things to work out normally for an "elevator" attached to an accelerating reference frame - after all, the elevator is just "accelerating" at some rate, albeit refined by hyperbolic motion, and relative signals should follow the usual (?) rule. OTOH, such an elevator moves through regions of increasing or decreasing proper acceleration, per Born motion. But the really big problem is this: if we let a small box free fall within the elevator, the Doppler shifts from one end to the other won't cancel out, since they are asymmetrical. (That is, the increments of velocity from falling will not cancel out the asymmetrical net shifts within the elevator.) Another problem I've thought of about the EP: if light is sent obliquely from one part of a system in hyperbolic motion towards a higher region, it should take very long to arrive, and be subject to great motional Doppler shift - more than the amount appropriate to the equivalent potential change. |
#2
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![]() "Neil" wrote in message ... This is a curious problem. It looks basic enough to have been considered before, but I find nothing related anywhere. Background: The gravitational Doppler shift formula is straightforward when the energy change is proportionately small. Tracking energy E, we take E_received = E0 (1 - g dot h/c^2). Following the equivalence principle, we expect this to apply in general to relative gravity fields, whether "real" or caused by relative motion, or even the *combination of both*. snip Just so there isn't any confusion about proper versus measured length, here's the corrected portion referencing proper elevator height L0: E_received = E0 [1 - gamma*g L0 (1 + v/c)/c^2] Since the g' felt in K' is multiplied by gamma (check with transformations), the actual discrepancy versus relative g' is E_received = E0 [1 - g' L0 (1 + v/c)/c^2]. This seems like it would violate energy conservation, since the photon's energy change does not correspond to the work doing moving the mass-energy in the local g', but remember that when we move the elevator, the impulse from photon emission and reception must be accounted for. |
#3
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"Neil" wrote in message ...
This is a curious problem. It looks basic enough to have been considered before, but I find nothing related anywhere. Background: The gravitational Doppler shift formula is straightforward when the energy change is proportionately small. Tracking energy E, we take E_received = E0 (1 - g dot h/c^2). Following the equivalence principle, we expect this to apply in general to relative gravity fields, whether "real" or caused by relative motion, or even the *combination of both*. However, if you directly work out the change in relative energy for a photon traversing a frame moving at constant velocity in a g-field (ie, elevator), the result is not as given by the Eq. applied to local values of g and h for the elevator K'. This problem would not appear in Newtonian physics. (Let us know if you've heard of anything like this problem, especially with references.) I think this effect is testable with current equipment. Critique is fine, but it won't help unless you show your work. I think sci.astro is appropriate given the GR crowd, and the problem could have cosmological consequences. The point can be summarized as: First, assume you are in a "rest frame" K which follows the normal rules given above, and that once light is emitted it follows consistent rules. Then, consider a moving frame and track the paths and energy changes of photons emitted from one place to another, combining the effects of both gravity and local relative velocity. Photons emitted from the trailing end P1 of the elevator travel farther along the "rest" frame of the gravity field to catch up, and thus suffer greater gravitational Doppler shift in transit than would normally apply with that elevator height. Photons emitted from the leading end P2 of the elevator travel less rest-frame distance than if the elevator were at rest, and show less GD shift. Since the velocities of the ends are locally the same, the velocity part of Doppler shift (which acts at the moments of emission and reception) cancels out, leaving the travel-based discrepancy. (Putting the math simply: the emission and reception Doppler formulae give inverse values, leaving the proportional change from gravitational effect to be solely that determined by the distance the photon travels in K.) Consider also reflecting a photon from the other end of the elevator: the photon is received at a different potential (in K) than when emitted, but combined velocity effects all cancel out. Therefore, it must show a net energy change, which wouldn't happen in "normal" gravity fields. It perplexes me too, but this is the result of directly working things out. I'll work it out in more detail, but using the simple case of light going parallel to g. If we combine Lorentz contraction (which does locally apply in a g-field) and "catch-up" calculations, we get the following for the values of h as actually moved in K, in terms of proper elevator height L0 and its velocity v, which is locally consistent: h = gamma*L0 (1 + v/c), with v signed negative when sent from a leading end and L0 signed negative when light moves down. Things may get more complicated when v approaches c, but at low v it is clear that the top of the elevator will move very nearly this extra margin before receiving a photon, etc. Hence, when we plug this formula into the GD shift we get E_received = E0 [1 - gamma*g L (1 + v/c)/c^2] Since the g' felt in K' is multiplied by gamma (check with transformations), the actual discrepancy versus relative g' is E_received = E0 [1 - g' L (1 + v/c)/c^2]. This seems like it would violate energy conservation, since the photon's energy change does not correspond to the work doing moving the mass-energy in the local g', but remember that when we move the elevator, the impulse from photon emission and reception must be accounted for. This problem raises questions about the equivalence principle also, since we'd expect things to work out normally for an "elevator" attached to an accelerating reference frame - after all, the elevator is just "accelerating" at some rate, albeit refined by hyperbolic motion, and relative signals should follow the usual (?) rule. OTOH, such an elevator moves through regions of increasing or decreasing proper acceleration, per Born motion. But the really big problem is this: if we let a small box free fall within the elevator, the Doppler shifts from one end to the other won't cancel out, since they are asymmetrical. (That is, the increments of velocity from falling will not cancel out the asymmetrical net shifts within the elevator.) Another problem I've thought of about the EP: if light is sent obliquely from one part of a system in hyperbolic motion towards a higher region, it should take very long to arrive, and be subject to great motional Doppler shift - more than the amount appropriate to the equivalent potential change. In any situation where velocity (direction) is ARBITRARILY given +/-, you may come across this contradiction, as velocity, force etc are ALWAYS +. They may be 'less than', but not reliant on direction for sign, and NOT 0. The Lorentz Transforms, SRelativity therefore BS...... Jim G |
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