No redshift in a freely falling frame

[stevendaryl]

The answer IS correct.

But your reasoning was incorrect.

 

[Austin0]

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.

What do you mean by this?
There would not seem to be a Rindler system covering both rockets. Such a system is predicated on an acceleration differential which is absent here.
If you extend a virtual Rindler frame from the rear rocket to the front rocket then the front rocket would actually be running slow relative to the colocated Rindler observer/clock , yes?
Are you perhaps basing your assumption on observed Doppler shift?

 

[stevendaryl]

But your reasoning was incorrect.

Let me complete the argument here, if anyone is actually reading. In order to talk about relative clock rates for distant clocks in a coordinate-independent fashion, you have to have some basis for choosing corresponding events: You pick two events e₁ at one clock, and a corresponding event e₂ at the second clock, an event e₃ at the first clock, and an event e₄ at the second clock. Then for these 4 events, one can compute a ratio of invariant intervals:

d\tau₁/d\tau₂

where d\tau₁ is the invariant interval between e₁ and e₃, and d\tau₂ is the invariant interval between e₂ and e₄. You obviously get different answers for the ratio, depending on which 4 events you choose for the comparison.

Criterion 1: equal coordinate times
There is one way of choosing the 4 events, which is to choose them according to coordinate time: Choose e₂ and e₄ so that the coordinate time difference, dt = t₄ - t₂ is equal to the coordinate time difference between e₁ and e₃, dt = t₃ - t₁. That is a coordinate-dependent criterion, and that gives you a coordinate-dependent notion of relative clock rates. Which is actually fine, because relative clock rates IS a coordinate-dependent quantity.

Criterion 2: Null geodesics
Another way of choosing the 4 events is to use null geodesics (a null geodesic is the path taken by a light signal): Choose e₂ and e₄ such that there is a null geodesic connecting e₁ and e₂, and similarly a null geodesic connecting e₃ and e₄. This criterion is what you care about if you are going to experimentally test gravitational redshift.

Now, to complete the argument, you need to use the fact that for a coordinate system such that (1) the components of the metric are static (independent of the time coordinate), and (2) the two clocks are stationary (none of the spatial coordinates are changing), the two criteria are the same. That is, if two light signals are sent from one location at a time dt apart, then they will arrive at the destination at a time dt apart.

Since the Schwarzschild coordinates do have a static metric components, criterion 1 is good enough.

 

[GAsahi]

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\tau² = X² dT² - dX²

So for clocks at rest in the X,T coordinates, we have:
d\tau = X dT

So the ratio of the rates is: d\tau₁/d\tau₂ = X₁/X₂

Conclusion: the "higher" clock (with greater X) runs faster.

This derivation is correct.

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\tau² = dt² - 1/c² dx²
= dt² (1 - v²/c²)
where v = dx/dt = the speed of the clock. So

d\tau = \sqrt{1-(v/c)^{2}} dtThe ratios of the rates in this coordinate system is given by:
d\tau₁/d\tau₂ = \sqrt{1-(v_{1}/c)^{2}}/\sqrt{1-(v_{2}/c)^{2}}

At t=0, v_{1} = v_{2} = 0. So the ratio starts off equal to 1, not X₁/X₂.

This part of the derivation is in error. If you did it correctly, you would have gotten that the correct result is \frac{d \tau_1}{d \tau_2}=\sqrt{\frac{1-v/c}{1+v/c}} where v 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 h=X_1-X_2. The bottom line is that there is always motion between the two ends of the rocket, so you cannot write

At t=0, v_{1} = v_{2} = 0. So the ratio starts off equal to 1, not X₁/X₂.

If the above were true, you would have found a way of disproving the principle of equivalence.

 

[Austin0]

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\tau² = X² dT² - dX²

So for clocks at rest in the X,T coordinates, we have:
d\tau = X dT

So the ratio of the rates is: d\tau₁/d\tau₂ = X₁/X₂

Conclusion: the "higher" clock (with greater X) runs faster.

This derivation is correct.

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\tau² = dt² - 1/c² dx²
= dt² (1 - v²/c²)
where v = dx/dt = the speed of the clock. So

d\tau = \sqrt{1-(v/c)^{2}} dt


The ratios of the rates in this coordinate system is given by:
d\tau₁/d\tau₂ = \sqrt{1-(v_{1}/c)^{2}}/\sqrt{1-(v_{2}/c)^{2}}

At t=0, v_{1} = v_{2} = 0. So the ratio starts off equal to 1, not X₁/X₂.

This part of the derivation is in error. If you did it correctly, you would have gotten that the correct result is \frac{d \tau_1}{d \tau_2}=\sqrt{\frac{1-v/c}{1+v/c}} where v 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 h=X_1-X_2. The bottom line is that there is always motion between the two ends of the rocket, so you cannot write

At t=0, v_{1} = v_{2} = 0. So the ratio starts off equal to 1, not X₁/X₂.

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?
Or comparably a ratio of rear dt/\gamma₁ front dt/\gamma₂ as measured in the launch frame.

More explicitly:If there are individual coordinate charts for the two clocks ,say from v=0 to v=0.9c then the integrated proper times for these worldlines must agree with the calculated proper times for these clocks using the Rindler metric Yes?

At t=0, v_{1} = v_{2} = 0. So the ratio starts off equal to 1, not X₁/X₂.

If the above were true, you would have found a way of disproving the principle of equivalence.

How could it not start out at At t=0, v_{1} = v_{2} = 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]

How could it not start out at At t=0, v_{1} = v_{2} = 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.

You missed the point that he got the wrong answer. So, his attempt at applying the second set of equations failed. The two approaches need to produce the SAME answer, otherwise he's found a way to disprove EPE. The measured ratio is X_1/X_2=\sqrt{\frac{1-v/c}{1+v/c}}, not 1 and, definitely NOT \sqrt{\frac{1-(v_1/c)^2}{1-(v_2/c)^2}}. THAT was the point.

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?

Clock hypothesis has very little , if any, to do with this problem.

 

[PeterDonis]

The measured ratio is X_1/X_2=\sqrt{\frac{1-v/c}{1+v/c}}, not 1 and, definitely NOT \sqrt{\frac{1-(v_1/c)^2}{1-(v_2/c)^2}}.

It looks to me like these two quantities refer to two different things. The first refers to Schwarzschild spacetime; the second refers to Rindler coordinates on Minkowski spacetime. The answers for those two cases will not be the same, because Schwarzschild spacetime is curved and Minkowski spacetime is flat.

 

[stevendaryl]

This part of the derivation is in error. If you did it correctly, you would have gotten that the correct result is \frac{d \tau_1}{d \tau_2}=\sqrt{\frac{1-v/c}{1+v/c}} where v is the instantaneous speed of the rocket containing the two clocks wrt the launcher frame.

That was exactly my point: You can't compute redshift by just computing d\tau₁ in terms of dx and dt, computing d\tau₂ in terms of dx and dt, and dividing them.

 

[stevendaryl]

You missed the point that he got the wrong answer.

That's exactly the point: your method of computing redshift gives the wrong answer unless two conditions are met:
(1) You are using a coordinate system in which the metric components are independent of time, and
(2) You are using a coordinate system in which the sender and the receiver of the light signals are both at rest in that coordinate system.

If those two conditions don't hold, then you can't simply compute d\tau₁ and d\tau₂ in terms of dx and dt, and get the right answer for redshift.[/QUOTE]

 

[stevendaryl]

It looks to me like these two quantities refer to two different things. The first refers to Schwarzschild spacetime; the second refers to Rindler coordinates on Minkowski spacetime. The answers for those two cases will not be the same, because Schwarzschild spacetime is curved and Minkowski spacetime is flat.

No, both are about flat spacetime. The difference is that
√(1-(v₁/c)²)/√(1-(v₂/c)²) is the ratio of the two clock rates, as measured in the "launch" frame, while √(1-v/c)/√(1+v/c) is the redshift formula for the case in which the "lower" clock sends a signal while at rest, and the signal is received by the "upper" clock when that clock is traveling at speed v. (Since the light signal takes time to propagate, the upper clock will have achieved a nonzero velocity while the light signal is in flight).

My point is that the redshift formula is NOT the same as the ratio of clock rates, except in very specific circumstances. Those circumstances actually hold for Rindler coordinates and for Schwarzschild coordinates, but they don't hold for arbitrary coordinates. The conditions for being able to equate "relative clock rates" with "redshift" are: (1) The metric tensor is independent of time, and (2) the sender and receiver are at rest in the coordinate system.

 

[GAsahi]

That's exactly the point: your method of computing redshift gives the wrong answer unless two conditions are met:
(1) You are using a coordinate system in which the metric components are independent of time, and

Correct.

(2) You are using a coordinate system in which the sender and the receiver of the light signals are both at rest in that coordinate system.

Incorrect. A simple disproof can be found in the way N.Ashby does the computations explaining the GPS functionality (see his paper in Living Reviews). The receiver and the emitter are NOT at rest wrt each other.

 

[GAsahi]

That was exactly my point: You can't compute redshift by just computing d\tau₁ in terms of dx and dt, computing d\tau₂ in terms of dx and dt, and dividing them.

This is standard textbook stuff, I have given you a couple of links that contradict your statement.

 

[stevendaryl]

The two approaches need to produce the SAME answer, otherwise he's found a way to disprove EPE.

It has nothing to do with the equivalence principle--the same two approaches can be used in Schwarzschild geometry or in flat spacetime, and you get the same two answers. Those answers are NOT the same, because they are asking different questions:

1. What is the ratio R_lower/R_upper of the rate of the upper clock to the rate of the lower clock? This ratio is a coordinate-dependent quantity. It has different values in different coordinate systems. That's true for flat spacetime and for the Schwarzschild geometry.
2. What is the ratio f_upper/f_lower of a light signal sent from the lower clock to the upper clock, where f_upper is the frequency of the signal when it is received, as measured by the upper observer, and f_lower is the frequency of the signal when it is sent, as measured by the lower observer. This ratio has the same value in every coordinate system, and is less than 1 (the frequency as received is lower than the frequency as sent). That's true for flat spacetime and for the Schwarzschild geometry.

The two ratios are only the same for a special coordinate system in which (1) the components of the metric tensor are time-independent, and (2) the two clocks are at rest in that coordinate system.

 

[GAsahi]

It looks to me like these two quantities refer to two different things. The first refers to Schwarzschild spacetime; the second refers to Rindler coordinates on Minkowski spacetime. The answers for those two cases will not be the same, because Schwarzschild spacetime is curved and Minkowski spacetime is flat.

Actually, IF you do the calculations correctly and IF you apply the EPE correctly, they ARE the same, with a very high degree of precision. EPE tells you that they must be the same. Attached please see the complete calculations.

 

[stevendaryl]

This is standard textbook stuff, I have given you a couple of links that contradict your statement.

You are very confused. What I pointed out is that there are two different ratios that can be computed: (1) the ratios of clock rates, and (2) the ratios of frequencies for a light signal sent from one observer to another. Those two ratios are only the same in the special case in which the coordinates used are such that the components of the metric tensor are time-independent, and the two observers are at rest in that coordinate system.

What in the textbook contradicts the above statements?

Note, that the two conditions are true for Schwarzschild coordinates (which are the coordinates usually used in Pound-Rebka experiments). They are also true for Rindler coordinates. But they are not true for arbitrary coordinates.

 

[GAsahi]

The two ratios are only the same for a special coordinate system in which (1) the components of the metric tensor are time-independent,

Yes.

and (2) the two clocks are at rest in that coordinate system.

No. I have already corrected you on this statement. IF it were true (it isn't) the GPS calculations would not work. Please see the N.Ashby paper in Living Reviews as reference.

 

[stevendaryl]

Actually, IF you do the calculations correctly and IF you apply the EPE correctly, they ARE the same, with a very high degree of precision. EPE tells you that they must be the same. Attached please see the complete calculations.

No, they are not the same, to a high degree of precision. They are different ratios: One is a ratio of clock rates, as measured in the "launch" frame. The other is the ratio of frequencies for a light signal. Those two ratios are NOT the same.

 

[GAsahi]

No, they are not the same, to a high degree of precision. They are different ratios: One is a ratio of clock rates, as measured in the "launch" frame. The other is the ratio of frequencies for a light signal. Those two ratios are NOT the same.

I appended my calculations that show they are the same (with a very high degree of precision).
Can you post your calculations that support your PoV?

 

[stevendaryl]

Actually, IF you do the calculations correctly and IF you apply the EPE correctly, they ARE the same, with a very high degree of precision. EPE tells you that they must be the same. Attached please see the complete calculations.

What your enclosed calculations show is that the ratio of clock rates (which is computed in the first calculation using Schwarzschild coordinates) gives the same answer as the redshift formula (computed in the second calculation using Doppler shift). I'm AGREEING with that. The redshift formula is a coordinate-independent quantity, which has the same value in any coordinate system (if it's done correctly). The ratio of clock rates is a coordinate-dependent quantity; it has DIFFERENT values in different coordinate systems. But--and I've already said this several times--if you use special coordinates in which (1) the metric is independent of time, and (2) the sender and receiver are at rest in that coordinate system, then for that particular coordinate system, the two answers are the same.

Your calculations are not contradicting those claims, they are illustrating them. If instead of using the Schwarzschild coordinates to compute relative clock rates, you had used a different coordinate system to compute relative clock rates, you would have gotten a different answer. I showed you that, by using inertial coordinates to compute relative clock rates for accelerating clocks. The ratios of the clock rates are 1 at time t=0, as computed in the inertial coordinates of the "launch" frame. That MUST be the case, because the clocks are initially at REST, and in inertial coordinates, the only relevant factors involved in clock rates are velocities. But the redshift formula does NOT a null result. If gives the same result whether you use inertial coordinates or Rindler coordinates.

 

[stevendaryl]

No. I have already corrected you on this statement. IF it were true (it isn't) the GPS calculations would not work. Please see the N.Ashby paper in Living Reviews as reference.

You are very confused. GPS calculations don't contradict what I have said. What I said is that redshift formula does not agree with the ratio of clock rates in an arbitrary coordinate system. That is certainly true. I gave you an explicit calculation proving it.

GPS calculations are done in a very specific coordinate system; most likely Schwarzschild coordinates, since that's the most convenient for an approximately spherically symmetrical case. In Schwarzschild coordinates, it IS the case that the redshift formula between two observers at rest (say, one at the bottom of a mountain, and one at the top of a mountain) will be equal to the ratio of clock rates.

(Of course, the real situation for GPS calculations is a lot more complicated, because Earth-based coordinates are a rotating coordinate system, relative to the Schwarzschild coordinates, and because sender and receiver may not be at rest, so we have to include Doppler effects as well as the Schwarzschild effects.)

What I've said is indisputably true. The ratio of clock rates is a coordinate-dependent quantity. Redshift between two observers is a coordinate-independent quantity. Those two statements are indisputably true.

 

[GAsahi]

What your enclosed calculations show is that the ratio of clock rates (which is computed in the first calculation using Schwarzschild coordinates) gives the same answer as the redshift formula (computed in the second calculation using Doppler shift). I'm AGREEING with that.

Then what are you splitting hairs about?

But--and I've already said this several times--if you use special coordinates in which (1) the metric is independent of time, and (2) the sender and receiver are at rest in that coordinate system, then for that particular coordinate system, the two answers are the same.

Your second condition is false , as shown by the way the GPS calculations are being done. I have already pointed this to you three times. The emitter and the receiver are in motion wrt each other, yet the calculations hold.

Your calculations are not contradicting those claims, they are illustrating them. If instead of using the Schwarzschild coordinates to compute relative clock rates, you had used a different coordinate system to compute relative clock rates, you would have gotten a different answer. I showed you that, by using inertial coordinates to compute relative clock rates for accelerating clocks.

It is not clear what mistake you made but I get the SAME result through both methods. If your claims were true the GPS calculations would fail.

 

[stevendaryl]

Incorrect. A simple disproof can be found in the way N.Ashby does the computations explaining the GPS functionality (see his paper in Living Reviews). The receiver and the emitter are NOT at rest wrt each other.

When they are NOT at rest wrt each other, the redshift formula is NOT the same as the ratio of the clock rates. The redshift differs from this rate by a Doppler correction. They explicitly say that in the Wikipedia article about Pound-Rebka:

Special Relativity predicts a Doppler redshift of :

f_r=\sqrt{\frac{1-v/c}{1+v/c}}f_e

On the other hand, General Relativity predicts a gravitational blueshift of:

f_r=\sqrt{\frac{1-\dfrac{2GM}{(R+h)c^2}}{1-\dfrac{2GM}{Rc^2}}}f_e

The detector at the bottom sees a superposition of the two effects.

 

[GAsahi]

GPS calculations are done in a very specific coordinate system; most likely Schwarzschild coordinates, since that's the most convenient for an approximately spherically symmetrical case. In Schwarzschild coordinates, it IS the case that the redshift formula between two observers at rest (say, one at the bottom of a mountain, and one at the top of a mountain) will be equal to the ratio of clock rates.

The emitter and receiver are NOT at rest wrt each other. The calculations done using Schwarzschild coordinates are confirmed by practice. So, your second claim is false.

 

[GAsahi]

When they are NOT at rest wrt each other, the redshift formula is NOT the same as the ratio of the clock rates. The redshift differs from this rate by a Doppler correction. They explicitly say that in the Wikipedia article about Pound-Rebka:

Special Relativity predicts a Doppler redshift of :

f_r=\sqrt{\frac{1-v/c}{1+v/c}}f_e

On the other hand, General Relativity predicts a gravitational blueshift of:

f_r=\sqrt{\frac{1-\dfrac{2GM}{(R+h)c^2}}{1-\dfrac{2GM}{Rc^2}}}f_e

The detector at the bottom sees a superposition of the two effects.

You misread the wiki paper: it tells you that the way to measure the gravitational effect is by cancelling it with the appropriate amount of Doppler effect by moving the source wrt the detector at the appropriate speed.

 

[stevendaryl]

Then what are you splitting hairs about?

Because you are making a serious mistake in confusing two different things:
(1) The ratio of clock rates, and (2) the redshift formula. They are not the same, except in special circumstances.

Your second condition is false , as shown by the way the GPS calculations are being done. I have already pointed this to you three times. The emitter and the receiver are in motion wrt each other, yet the calculations hold.

No, they don't. If the emitter and receiver are in motion, then the redshift formula has to be adjusted to include both position-dependent and velocity-dependent effects.

It is not clear what mistake you made

I didn't make a mistake, you did.

... but I get the SAME result through both methods. If your claims were true the GPS calculations would fail.

You didn't compute d\tau in terms of dt and dr for any coordinate system other than Schwarzschild. My claim is that if you had used a different coordinate system to compute d\tau for the two clocks, and taken the ratio, you would have gotten a different answer than you get for Schwarzschild.

For you to say "if your claims were true the GPS calculations would fail" makes no sense, because what I'm saying AGREES with what you are saying when Schwarzschild coordinates are used to compute d\tau. Since you haven't attempted to compute d\tau for any other coordinate system, the point of disagreement hasn't come up.

Well, it actually has come up, in the Rindler case, but you wisely declined to offer a calculation of d\tau in that case.

 

[stevendaryl]

You misread the wiki paper: it tells you that the way to measure the gravitational effect is by cancelling it with the appropriate amount of Doppler effect by moving the source wrt the detector at the appropriate speed.

Why do you think that that says something different from what I'm saying? I'm saying that if the two detectors are in motion relative to one another, Doppler shift must be included in the redshift calculation. That's clearly true. They say it right there in the article.

It explicitly says: "The detector at the bottom sees a superposition of the two effects",
where the two effects are position-dependent time dilation, and Doppler shift.

 

[stevendaryl]

The emitter and receiver are NOT at rest wrt each other. The calculations done using Schwarzschild coordinates are confirmed by practice. So, your second claim is false.

If the detector and the emitter are not at rest relative to each other (as measured in Schwarzschild coordinates) then the pure position-dependent gravitational time dilation must be corrected by an additional Doppler term. Are you disputing that? That's very bizarre. Think about it: suppose that the receiver and the sender are at the SAME height. Then the redshift is purely due to Doppler.

You CAN'T use the gravitational time dilation to compute redshift without including Doppler, except in the special case in which the sender and receiver are at rest (so that the Doppler effect is zero). You're not seriously disputing that, are you?

 

[GAsahi]

Because you are making a serious mistake in confusing two different things:
(1) The ratio of clock rates, and (2) the redshift formula. They are not the same, except in special circumstances.

Repeating the same error ad nauseaum doesn't make it right. Your so-called "counter-example" has the source and the emitter at rest wrt each other.
You are desperately trying to prove that the method does not apply when the emitter and the detector are moving wrt each other (you changed the goal posts when I showed you that the method works when there is no relative motion). The GPS calculations , as posted by Ashby, disprove your statement.

So, you have a "counter-example" that does not apply and your statements are contradicted by mainstream application of Schwarzschild coordinates to explaining the GPS functionality. You are 0 for 2.

 

[PeterDonis]

No, both are about flat spacetime.

Hm, ok, I need to go back and read your original posts more carefully. However, I'm not sure GAsahi is talking about flat spacetime (but maybe I need to go back and read his original posts more carefully too).

My point is that the redshift formula is NOT the same as the ratio of clock rates, except in very specific circumstances. Those circumstances actually hold for Rindler coordinates and for Schwarzschild coordinates, but they don't hold for arbitrary coordinates.

This is not correct as you state it; the circumstances are not coordinate-dependent. See below.

The conditions for being able to equate "relative clock rates" with "redshift" are: (1) The metric tensor is independent of time, and (2) the sender and receiver are at rest in the coordinate system.

It would be better if you stated these conditions in coordinate-free terms, which can be done:

(1) The spacetime has a timelike Killing vector field;

(2) The sender and receiver's worldlines are both orbits of the timelike Killing vector field.

That should make it clear that the conditions you are talking about depend on particular properties of the spacetime and the worldlines, but *not* on coordinates; the mathematical description of the conditions looks simpler in Schwarzschild coordinates (or Rindler in flat spacetime), but that doesn't mean it's only "true in" those coordinates.

 

[GAsahi]

If the detector and the emitter are not at rest relative to each other (as measured in Schwarzschild coordinates) then the pure position-dependent gravitational time dilation must be corrected by an additional Doppler term. Are you disputing that? That's very bizarre. Think about it: suppose that the receiver and the sender are at the SAME height. Then the redshift is purely due to Doppler.

You CAN'T use the gravitational time dilation to compute redshift without including Doppler, except in the special case in which the sender and receiver are at rest (so that the Doppler effect is zero). You're not seriously disputing that, are you?

I showed you how to do the calculations using the Schwarzschild solution for the case of relative motion between source and detector. You do not need any "additional Doppler term", the answer is fully contained in the Schwarzschild solution. You seem to have this bee under your bonnet that you can only use the Schwarzschild solution when the source and the detector are stationary.