The equivalence principle and blueshift

[DanMP]

In another thread, where twin paradox was discussed, Markus Hanke said:

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The equivalence principle tells us that uniform acceleration is locally equivalent to a uniform gravitational field; differently put, the accelerated twin sits at a different gravitational potential, which implies frequency shift.

I asked:

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So, here, on the Earth surface, we also see the light coming from stars straight above us "blueshifting" as long as our accelerometer shows that we are "accelerating" upward?

The first answer, from Mordred, was:

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As light climbs in and out of a gravity well it will blue shift or redshift. For example an outside observer looking at infalling material at the EH of a blackhole will see infinite redshift but an observer at the EH will see infinite blue shift. This is due to gravitational redshift 

True, but he didn't understand my question.

The second answer, from Markus, was:

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Technically yes, there will be some amount of frequency shift, though in practice the effect is quite small for a weak field such as the Earth’s.

Again, not what I asked, so I'll try again here.

The traveling twin, after the turnaround, accelerates toward Earth/Sun and he immediately see the light from the Sun blueshifting. As long as he accelerates, the light is blueshifted again and again, due to the increase in speed towards the source of the light (the Sun). So, as long as his accelerometer indicates an acceleration towards the Sun, the light of the Sun is blueshifted again and again.

On the Earth surface, say on the South pole, in summer, we also have the accelerometer indicating an acceleration towards the Sun, but the blueshift is not increasing, like in the traveling twin case. So the equivalence principle seems not quite relevant in explaining the blueshift that the travelling twin experiences. If I'm wrong, please explain me how/why I'm wrong.

Edited April 25 by DanMP

[Genady]

I think you can figure it out yourself. Hint: compare with tidal effects, which exist on Earth but not on an accelerating spacecraft.

Edited April 25 by Genady

[KJW]

In the case of the twin paradox, the redshift and blueshift are due to the relative velocity between the source and receiver. In the case of gravitational redshift and blueshift, equivalent to accelerational redshift and blueshift, there is no relative velocity between the source and receiver. The redshift and blueshift is due to the acceleration itself, and depends on the displacement between source and receiver.

If one examines the metric of an accelerated frame of reference, sources that are at rest and above the receiver (note that the acceleration is directed upward) will be blueshifted, while sources that are at rest and below the receiver will be redshifted.

It is worth noting that the source that is at rest below the receiver has a greater acceleration than the receiver, and that the source that is at rest above the receiver has a lesser acceleration than the receiver. However, this does not determine the redshift or blueshift, and it might not apply to gravitation.

 

Edited April 25 by KJW

[md65536]

2 hours ago, KJW said:

In the case of gravitational redshift and blueshift, equivalent to accelerational redshift and blueshift, there is no relative velocity between the source and receiver. The redshift and blueshift is due to the acceleration itself, and depends on the displacement between source and receiver.

If the equivalence principle is used, then there should be an equivalent relative velocity between source and receiver.

Consider 2 sources falling into a black hole, and only one of them realizes it and accelerates outward so that it can remain stationary relative to the EH. Near the EH, it would need to have a velocity approaching c relative to the free-falling source, so there would be a Doppler shift between the two. The sources would have to appear differently, and external sources would have to appear different to them, depending if they're stationary or falling.

 

Consider a rocket accelerating upward, so that the bottom of the rocket is equivalent to being deeper in a gravitational well relative to the top. Light from the top is blue-shifted when seen at the bottom, but the top and bottom remain at relative rest. However, consider two sources at the top of the rocket, both emitting a single pulse of light. One of the sources is fixed to the rocket and accelerating, and the other is inertial but set up so that it is momentarily at rest with the other source at the moment the pulse is emitted. Both pulses should be blue-shifted the same amount when seen by the bottom of the rocket, even though the rocket will have a relative velocity with one of the sources when it is seen. Or to put it another way, since the rocket is accelerating and light takes time to cross the distance of the rocket, the velocity of the receiver at the moment of reception will be different than the velocity of the source at emission. The blueshift can be entirely attributed to this difference in velocity, based on reasoning when source and emitter are replaced with equivalent but inertial particles.

 

[KJW]

All redshifts and blueshifts ultimately have the same cause: Consider some arbitrary spacetime in which there is an arbitrary trajectory of an emitter and an arbitrary trajectory of a receiver. From the emitter, consider two infinitesimally separated future-directed light-like geodesics that eventually intersect the trajectory of the receiver. Then the ratio of the proper time along the trajectory of the emitter between the two light-like geodesics, and the proper time along the trajectory of the receiver between the two light-like geodesics, is the Doppler ratio (redshift or blueshift) that the receiver observes at the instant of receiving the two light-like geodesics from the emitter.

From the above, one may consider more special cases where the cause of the redshift and blueshifts can be differentiated between relative velocity, acceleration, gravitation, or cosmological, as well as combinations of these. For example, constant acceleration in the Minkowskian metric can be considered in terms of relative velocity, whereas the same constant acceleration in its own frame of reference can be considered in terms of pure acceleration. In other words, the change in metric, even though the physics has not changed, leads to a change in interpretation. But in all cases, the above applies. And note that the above has been expressed entirely in terms of invariants.

 

Edited April 25 by KJW

[Mordred]

Just to add for acceleration involving change in direction will involve transverse redshift. 

Just to add some useful relations more for the benefit of any readers not familiar with the types of redshift.

\[\frac{\Delta_f}{f} = \frac{\lambda}{\lambda_o} = \frac{v}{c}=\frac{E_o}{E}=1+\frac{hc}{\lambda_o} \frac{\lambda}{hc}\]

Doppler shift

\[z=\frac{v}{c}\]

Relativistic Doppler redshift 

\[1+z=(1+\frac{v}{c})\gamma\]

Transverse redshift

\[1+z=\frac{1+v Cos\theta/c}{\sqrt{1-v^2/c^2}}\]

If \(\theta=0 \) degrees this reduces to 

\[1+z=\sqrt\frac{1+v/c}{1-v/c}\]

At right angles this gives a redshift even though the emitter is not moving away from the observer

\[1+z=\frac{1}{\sqrt{1-v^2/c^2}}\]

From this we can see the constant velocity twin will have a transverse Doppler even though the velocity is constant.

The acceleration as per change in velocity is straight forward with the above equations as the redshift/blueshift will continously change with the change in velocity term.

The equations in this link will help better understand the equivalence principle in regards to gravity wells such as a planet

https://en.m.wikipedia.org/wiki/Pound–Rebka_experiment

The non relativistic form  being

\[\acute{f}=f(1+\frac{gh}{c^2})\]

 

 

 

 

Edited April 26 by Mordred

[DanMP]

After I left yesterday, I realized that Markus Hanke answer regarding the blueshift:

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The equivalence principle tells us that uniform acceleration is locally equivalent to a uniform gravitational field; differently put, the accelerated twin sits at a different gravitational potential, which implies frequency shift.

ignored that the source of light, the Sun, is not comoving with the travelling twin. For a light source situated in the accelerating spaceship, in front of the twin, there would be the blueshift that Markus Hanke mentioned, but if the source of light, the Sun in this case, is in front of the spaceship, outside, not comoving, the blueshift would continue to grow as the velocity of the ship increases.

My point is that if an accelerometer indicates an acceleration and the blueshift of the stars in the direction of acceleration is not increasing, the acceleration is not real, it is not associated with an increase in velocity. In this way we can distinguish between uniform acceleration and uniform gravitational field.

[KJW]

5 hours ago, DanMP said:

My point is that if an accelerometer indicates an acceleration and the blueshift of the stars in the direction of acceleration is not increasing, the acceleration is not real, it is not associated with an increase in velocity. In this way we can distinguish between uniform acceleration and uniform gravitational field.

Note that Markus Hanke said that "uniform acceleration is locally equivalent to a uniform gravitational field" [my bold]. The distant stars are not part of a local measurement.

 

[Markus Hanke]

15 hours ago, DanMP said:

In this way we can distinguish between uniform acceleration and uniform gravitational field.

As KJW has pointed out, the equivalence principle is a purely local statement, meaning it applies only to small spacetime regions - meaning small volumes over short periods of time. Within such regions, there’s no purely local experiment you can perform to distinguish between the two.

For the twin scenario, the twin can at every individual instant (or short enough time period) be considered to be subject to a uniform gravitational field, where the gravitational potential may vary from instant to instant. This kind of foliation procedure produces the correct results - though I still don’t think it’s necessarily the best way to analyse the twin scenario.

[DanMP]

On 4/26/2024 at 9:49 PM, KJW said:

Note that Markus Hanke said that "uniform acceleration is locally equivalent to a uniform gravitational field" [my bold]. The distant stars are not part of a local measurement.

Ok, but the context was the twin paradox and the blueshift observed at the turnaround. The blueshifted signal originated from Earth, clearly not local.

 

On 4/27/2024 at 8:30 AM, Markus Hanke said:

This kind of foliation procedure produces the correct results - though I still don’t think it’s necessarily the best way to analyse the twin scenario.

Ok, so why did you offer the equivalence principle as an explanation for the blueshift there, in the twin scenario discussion?

 

Anyway, the mentioned/linked discussion was closed and I don't intend to continue/reopen it here. All I wanted here was to make this observation:

On 4/26/2024 at 4:39 PM, DanMP said:

My point is that if an accelerometer indicates an acceleration and the blueshift of the stars in the direction of acceleration is not increasing, the acceleration is not real, it is not associated with an increase in velocity. In this way we can distinguish between uniform acceleration and uniform gravitational field.

 

[swansont]

10 minutes ago, DanMP said:

Ok, but the context was the twin paradox and the blueshift observed at the turnaround. The blueshifted signal originated from Earth, clearly not local.

But that’s a velocity-induced blueshift. The acceleration is incidental; you’ll see that blueshift regardless of the acceleration details. 

[DanMP]

6 minutes ago, swansont said:

But that’s a velocity-induced blueshift. The acceleration is incidental; you’ll see that blueshift regardless of the acceleration details. 

Yes, that's why I asked Markus Hanke:

18 minutes ago, DanMP said:

Ok, so why did you offer the equivalence principle as an explanation for the blueshift there, in the twin scenario discussion?

 

Edited April 29 by DanMP

[Markus Hanke]

19 hours ago, DanMP said:

Ok, so why did you offer the equivalence principle as an explanation for the blueshift there, in the twin scenario discussion?

Because the author of that thread explicitly asked about frequency shift, so I offered one particular way to look at that.

Personally though I would altogether avoid any analysis of what happens at every instant in different frames, and simply compare the lengths of the two world lines. That works irrespective of the details of how the twins move, and irrespective of what spacetime they find themselves in.