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vin300
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Why is there no redshift in a freely falling frame? The photon in a freely falling frame also rises in the gravitational field, so isn't it supposed to be redshifted?
Naty1 said:Does the PeterDonis description also apply to the deBroglie wavelength of a free falling matter particle?
I'm thinking of the Tamara Davis article I posted where she notes such a deBroglie wavelength of a matter particle redshifts in the same proportion in cosmological expansion as the wavelength [redshift] of a photon.
I'm wondering if that analogy applies to gravitational redshift as well.
Naty1 said:I'm having trouble understanding " but the observer will have picked up just enough downward velocity so that when the observer receives the photon..."
In what sense is the 'at rest' observer accelerating downward??
TSny said:As you can see from PeterDonis' excellent answer, "redshift" (wavelength) is not a property belonging to a photon itself. It is a joint property of a photon and an observer (detector).
vin300 said:Can it be shown mathematically?
Nowadays the Fermi normal coordinates are usually - although improperly - called Fermi coordinates. In exper-
imental gravitation, Fermi normal coordinates are a powerful tool used to describe various experiments: since the
Fermi normal coordinates are Minkowskian to first order, the equations of physics in a Fermi normal frame are the
ones of special relativity, plus corrections of higher order in the Fermi normal coordinates, therefore accounting for
the gravitational field and its coupling to the inertial effects. Additionally, for small velocities v compared to light
velocity c, the Fermi normal coordinates can be assimilated to the zeroth order in (v/c) to classical Galilean coordi-
nates. They can be used to describe an apparatus in a “Newtonian” way (e.g. [1, 3, 8, 10]), or to interpret the outcome
of an experiment (e.g. [11] and comment [21], [5, 6, 15, 17]). In these approaches, the Fermi normal coordinates are
considered to have a physical meaning, coming from the principle of equivalence (see e.g. [18]), and an operational
meaning: the Fermi normal frame can be realized with an ideal clock and a non extensible thread [29].
In the sentence you quoted, "observer" refers to a freely falling observer. I was trying to describe how an observer static in the field would view the observations made by a freely falling observer.
In the case of a photon in free space, its velocity is constant. In the case of a massive particle, things are more complicated. The particle's velocity is not constant, since a Lorentz transformation changes
...I'm not sure it can be described as simply as the effect on a photon can be.
oops now I am not so sure:...I'm not sure it can be described as simply as the effect on a photon can be.
I think you are right:
Neither photons nor even massive objects change in frequency or energy
when moving in a static gravitational field as observed by anyone observer. In Newtonian terms this is because the combined effects of changes in kinetic energy (from motion) plus potential energy (from time dilation) result in constant total energy.
yuiop said:If the source is stationary relative to the gravitational field, then a stationary observer lower down in the gravitational well, will see the light as blue shifted and a free falling observer lower down will see the same light as red shifted.
Naty1 said:Upon reconsideration I think the above from Scott is incorrect regarding frequency: frequency corresponds to the KE of light so it should change as a photon moves in a non uniform static gravitational field.
vin300 said:I've found an explanation that violates the equivalence principle. An observer in a uniform gravitational field sees objects accelerating downwards at a constant g. An observer with a constant upward acceleration g also sees the same thing, but there's something that's not equivalent in the two cases. An observer in a gravitational field says that a falling object has an unchaging total energy, all the gain in its kinetic energy is due to equivalent loss of potential energy, but the accelerating observer doesn't say that its potential energy is decreasing. The man in the field says the energy of the object doesn't change, but the accelerating one says that the energy changes at a constant rate.
GAsahi said:It is considerably more complicated than the above. The first half of your statement is correct, the second one is not. Proof
Start with the (reduced) Schwarzschild metric:
[tex]d \tau^2=(1-r_s/r)dt^2-dr^2/(1-r_s/r)[/tex]
1. If the two observer and the source are stationary, then [itex]dr=0[/itex] at radial coordinates [itex]r_1<r_2[/itex] , you can write:
[tex]d \tau_1^2=(1-r_s/r_1)dt^2[/tex]
[tex]d \tau_2^2=(1-r_s/r_2)dt^2[/tex]
It follows that :
[tex]\frac{d \tau_1}{d \tau_2}=\sqrt {\frac{1-r_s/r_1}{1-r_s/r_2}}[/tex]
Since [itex]r_1<r_2[/itex] it follows
[tex]d \tau_1< d \tau_2[/tex]
i.e [tex]f_1>f_2[/tex] (blueshift)
2. If the source is moving, things get more complicated.
[tex]d \tau_1^2=(1-r_s/r_1)dt^2[/tex]
[tex]d \tau_2^2=(1-r_s/r)dt^2-dr^2/(1-r_s/r)=(1-r_s/r)dt^2 (1-\frac{(dr/dt)^2}{(1-r_s/r)^2})[/tex]
Therefore:
[tex]\frac{d \tau_1^2}{d \tau_2^2}=\frac{1-r_s/r_1}{1-r_s/r}\frac{1}{1-\frac{v^2}{(1-r_s/r)^2}}[/tex]
The second fraction is always larger than 1.
The first fraction is sometimes greater than 1 (for [itex]r_1>r[/itex]) or smaller than 1, (for [itex]r_1<r[/itex]), so you can get either blue or redshift (you don't always get redshift).
For [itex]v^2=(1-r_s/r)(r_s/r_1-r_s/r)[/itex] there is no shift whatsoever.
stevendaryl said:I think that your answers are correct, but I think that the computation is not quite correct. The reason why is because what you have calculated is the ratio of clock rates as measured using Schwarzschild coordinates. Relative clock rates for spatially separated clocks is a coordinate-dependent quantity; a different coordinate system might give a different answer.
The quantity that is directly observable, and is independent of coordinate systems, is this:
Let a photon be emitted from one observer. Let f1 be its frequency, as measured by that observer. Let the photon travel to the other observer. Let f2 be the frequency of that same photon, as measured by the second observer. Then the measured redshift or blueshift would be f2/f1.
I don't know right off the bat if that gives the same answer as the ratio you computed, or not, but conceptually, they are different.
GAsahi said:What I calculated is the ratio of proper periods. This is known to be coordinate independent.
stevendaryl said:No, in general, the ratio that you computed is not coordinate-independent. How are you defining a "proper period"?
What you computed was equivalent to the following:
[*]Let e1 be some event at the first observer (the one at radius r1).
[*]Let e2 be some event at the second observer (the one at radius r2) that is simultaneous with e1.
[*]Let e3 be some event at the first observer at a time δt later than e1.
[*]Let e4 be some event at the second observer at a time δt later than e2.
[*]Let d[itex]\tau[/itex]1 be the invariant interval between e1 and e3.
[*]Let d[itex]\tau[/itex]2 be the invariant interval between e2 and e4.
[*] Then define the relative rates to be the ratio d[itex]\tau[/itex]1/d[itex]\tau[/itex]2
stevendaryl said:The easiest way to see this is in the case of two identically accelerating rockets in flat spacetime, each equipped with an onboard clock. Using an inertial coordinate system, the ratio of the rates will be 1, because the time dilation effects will be the same for both clocks. Using the Rindler coordinate system in which the rear rocket is at rest, the front clock will be seen to be running faster than the rear clock.
GAsahi said:You got this part right. Now, given (by your own admission) that d[itex]\tau[/itex]1 and d[itex]\tau[/itex]2 are invariant, it follows that their ratio d[itex]\tau[/itex]1/d[itex]\tau[/itex]2 is also invariant.
stevendaryl said:Let me do an explicit calculation to prove my point.
In Rindler coordinates (X,T), we have two clocks, one at X = X1, and one at X2. The Rindler interval is:
d[itex]\tau[/itex]2 = X2 dT2 - dX2
So for clocks at rest in the X,T coordinates, we have:
d[itex]\tau[/itex] = X dT
So the ratio of the rates is: d[itex]\tau[/itex]1/d[itex]\tau[/itex]2 = X1/X2
Conclusion: the "higher" clock (with greater X) runs faster.
Now, do the same calculation in the coordinate system (x,t) related to (X,T) through:
x = X cosh(gT)
t = X/c sinh(gT)
So d[itex]\tau[/itex]2 = dt2 - 1/c2 dx2
= dt2 (1 - v2/c2)
where v = dx/dt = the speed of the clock. So
d[itex]\tau[/itex] = [itex]\sqrt{1-(v/c)^{2}}[/itex] dt
The ratios of the rates in this coordinate system is given by:
d[itex]\tau[/itex]1/d[itex]\tau[/itex]2 = [itex]\sqrt{1-(v_{1}/c)^{2}}[/itex]/[itex]\sqrt{1-(v_{2}/c)^{2}}[/itex]
At t=0, [itex]v_{1}[/itex] = [itex]v_{2}[/itex] = 0. So the ratio starts off equal to 1, not X1/X2.
stevendaryl said:What you wrote down was:
d[itex]\tau[/itex]1 = √(1-r/r1) dt
d[itex]\tau[/itex]2 = √(1-r/r2) dt
That corresponds to choosing the four events so that δt1 = δt2, where δt1 = the coordinate time between e1 and e3, and δt2 = the coordinate time between e2 and e4. That choice is coordinate-dependent. If you meant for a coordinate-independent choice, then what is your basis for choosing the events e1, e2, e3 and e4?
GAsahi said:It is really simple, IF you did your calculations correctly, you SHOULD have obtained the same result.
In fact, correct calculations are confirmed by experiment, in this case it is the Pound-Rebka experiment.
GAsahi said:OK, I understand what you are doing wrong. You totally misunderstand the problem statement (and, consequently, what I have been doing).
Observer1, located at [itex]r_1[/itex] measures on HIS clock, the interval [itex]d \tau_1[/itex].
Observer2 , located at [itex]r_2[/itex] measures remotely the SAME interval, on HIS clock (identical to the one used by Observer1), obtaining the value [itex] d \tau_2[/itex].
stevendaryl said:My original post said that your answer might be correct.
GAsahi said:The answer IS correct.
stevendaryl said:The easiest way to see this is in the case of two identically accelerating rockets in flat spacetime, each equipped with an onboard clock. Using an inertial coordinate system, the ratio of the rates will be 1, because the time dilation effects will be the same for both clocks. Using the Rindler coordinate system in which the rear rocket is at rest, the front clock will be seen to be running faster than the rear clock.
stevendaryl said:But your reasoning was incorrect.
stevendaryl said:Let me do an explicit calculation to prove my point.
In Rindler coordinates (X,T), we have two clocks, one at X = X1, and one at X2. The Rindler interval is:
d[itex]\tau[/itex]2 = X2 dT2 - dX2
So for clocks at rest in the X,T coordinates, we have:
d[itex]\tau[/itex] = X dT
So the ratio of the rates is: d[itex]\tau[/itex]1/d[itex]\tau[/itex]2 = X1/X2
Conclusion: the "higher" clock (with greater X) runs faster.
Now, do the same calculation in the coordinate system (x,t) related to (X,T) through:
x = X cosh(gT)
t = X/c sinh(gT)
So d[itex]\tau[/itex]2 = dt2 - 1/c2 dx2
= dt2 (1 - v2/c2)
where v = dx/dt = the speed of the clock. So
d[itex]\tau[/itex] = [itex]\sqrt{1-(v/c)^{2}}[/itex] dtThe ratios of the rates in this coordinate system is given by:
d[itex]\tau[/itex]1/d[itex]\tau[/itex]2 = [itex]\sqrt{1-(v_{1}/c)^{2}}[/itex]/[itex]\sqrt{1-(v_{2}/c)^{2}}[/itex]
At t=0, [itex]v_{1}[/itex] = [itex]v_{2}[/itex] = 0. So the ratio starts off equal to 1, not X1/X2.
At t=0, [itex]v_{1}[/itex] = [itex]v_{2}[/itex] = 0. So the ratio starts off equal to 1, not X1/X2.
stevendaryl said:Let me do an explicit calculation to prove my point.
In Rindler coordinates (X,T), we have two clocks, one at X = X1, and one at X2. The Rindler interval is:
d[itex]\tau[/itex]2 = X2 dT2 - dX2
So for clocks at rest in the X,T coordinates, we have:
d[itex]\tau[/itex] = X dT
So the ratio of the rates is: d[itex]\tau[/itex]1/d[itex]\tau[/itex]2 = X1/X2
Conclusion: the "higher" clock (with greater X) runs faster.
GAsahi said:This derivation is correct.
stevendaryl said:Now, do the same calculation in the coordinate system (x,t) related to (X,T) through:
x = X cosh(gT)
t = X/c sinh(gT)
So d[itex]\tau[/itex]2 = dt2 - 1/c2 dx2
= dt2 (1 - v2/c2)
where v = dx/dt = the speed of the clock. So
d[itex]\tau[/itex] = [itex]\sqrt{1-(v/c)^{2}}[/itex] dt
The ratios of the rates in this coordinate system is given by:
d[itex]\tau[/itex]1/d[itex]\tau[/itex]2 = [itex]\sqrt{1-(v_{1}/c)^{2}}[/itex]/[itex]\sqrt{1-(v_{2}/c)^{2}}[/itex]
At t=0, [itex]v_{1}[/itex] = [itex]v_{2}[/itex] = 0. So the ratio starts off equal to 1, not X1/X2.
GAsahi said:This part of the derivation is in error. If you did it correctly, you would have gotten that the correct result is [itex]\frac{d \tau_1}{d \tau_2}=\sqrt{\frac{1-v/c}{1+v/c}}[/itex] where [itex]v[/itex] is the instantaneous speed of the rocket containing the two clocks wrt the launcher frame.
I can get into all the details of why the above is the correct result but I won't , the way to get the correct result is not simply using the equations of hyperbolic motion, you can simply use the equivalence principle and to observe the Doppler effect on the frequency emitted at one end of the rocket and received at the other end, the two ends being separated by a distance [itex]h=X_1-X_2[/itex]. The bottom line is that there is always motion between the two ends of the rocket, so you cannot write
It would appear that if stevendaryl's calculation of relative rate as the relationship between instantaneous gammas is incorrect then a basic principle of SR falls. Specifically the Clock Hypothesis. Delta t' for either clock must be equal to an integration over that worldline interval based on instantaeous (infinitesimal) velocity gammas ,yes?At t=0, [itex]v_{1}[/itex] = [itex]v_{2}[/itex] = 0. So the ratio starts off equal to 1, not X1/X2.
At t=0, [itex]v_{1}[/itex] = [itex]v_{2}[/itex] = 0. So the ratio starts off equal to 1, not X1/X2.
How could it not start out at At t=0, [itex]v_{1}[/itex] = [itex]v_{2}[/itex] = 0 ? it does not instantaneously attain its final proper acceleration ,yes?? It would seem it would have to start out at 1 and over some finite time interval reach the relative ratio.GAsahi said:If the above were true, you would have found a way of disproving the principle of equivalence.