Can gravitational blue shift happen without gravitational time dilation?

[nutgeb]

First, a simple scenario to set our expectations:

Consider a "stationary" massless non-relativistic test particle in empty space. We'll call the particle "P". At a very large proper distance to the cosmological "west" of P there is a neutron star, at proper rest relative to P. Let's call it Star West. At the same proper distance to the "east" of P there is another identical neutron star, also at proper rest. We'll call it Star East.

Now we give a shove to both stars, radially inward toward P. Eventually their gravitational acceleration toward each other will become significant, and they will continue gaining velocity until they smash into each other and crush poor P between them.

An observer on Star West considers Star West to be stationary, P to be accelerating toward Star West, and Star East to be accelerating at an even higher rate toward Star West. (An observer on Star East sees the reverse occurring). From the Star West observer's local perspective, P is gaining kinetic energy as it freefalls toward Star West, and Star East is gaining twice as much kinetic energy.

The Star West observer concludes that the presence of Star East has no effect on how much local kinetic energy P gains. In other words, P's kinetic energy relative to Star West is not reduced by the fact that P is also accelerating toward, and gaining kinetic energy locally relative to, Star East. This happens because the two stars are in freefall toward each other.
. . . . . . . . . .

Now let's consider what happens when P is replaced with a relativistic photon:

Instead of having P start in the center, let's have Star East fire a single photon radially toward Star West at some point in time while the two stars are gravitationally accelerating toward each other.

When the photon arrives at Star West (let's say it arrives long before the two Stars crash together), the observer there will measure the photon to be gravitationally blueshifted. (For this discussion, we will divide out the component of SR Doppler blueshift so as to ignore it, and focus only on the component of gravitational blueshift). The photon is gravitationally blueshifted because, as observed in Star West's local frame, the photon experienced gravitational acceleration toward Star West all along its worldline. It didn't gain proper velocity from the acceleration, instead it became gravitationally blueshifted. And based on the prior experience with the nonrelativistic particle P, we know that any gravitational acceleration the photon experiences in the direction of Star East will not reduce the kinetic energy the photon gains toward Star West.

This is an interesting outcome. Normally, a photon that travels from the surface of one star and is observed on the surface of another identical star will not experience ANY net gravitational blueshift. It will become gravitationally redshifted as it climbs out of the gravitational well of the source star, and then will become gravitationally blueshifted by exactly the same amount as it falls into the gravitational well on the observer star.

Yet the blueshift occurs in our scenario because the two stars are in freefall toward each other. At the same time, there can be no gravitational time dilation as between Star East and Star West, because their masses are identical. So we have gravitational blueshift without any accompanying gravitational time dilation.

Yes?

 

[JesseM]

Consider a "stationary" massless non-relativistic test particle in empty space. We'll call the particle "P". At a very large proper distance to the cosmological "west" of P there is a neutron star, at proper rest relative to P. Let's call it Star West. At the same proper distance to the "east" of P there is another identical neutron star, also at proper rest. We'll call it Star East.

Now we give a shove to both stars, radially inward toward P. Eventually their gravitational acceleration toward each other will become significant, and they will continue gaining velocity until they smash into each other and crush poor P between them.

An observer on Star West considers Star West to be stationary, P to be accelerating toward Star West, and Star East to be accelerating at an even higher rate toward Star West. (An observer on Star East sees the reverse occurring). From the Star West observer's local perspective, P is gaining kinetic energy as it freefalls toward Star West, and Star East is gaining twice as much kinetic energy.

The Star West observer concludes that the presence of Star East has no effect on how much local kinetic energy P gains. In other words, P's kinetic energy relative to Star West is not reduced by the fact that P is also accelerating toward, and gaining kinetic energy locally relative to, Star East. This happens because the two stars are in freefall toward each other.

How are you defining "kinetic energy"? In GR we can use arbitrary global coordinate systems in curved spacetime, and kinetic energy is normally coordinate-dependent. I suppose you could talk about the kinetic energy in the locally inertial frame of an observer on the surface of a star at the moment P reaches that observer, but in any case the statement "The Star West observer concludes that the presence of Star East has no effect on how much local kinetic energy P gains" appears to be a non sequitur, nothing you've said in the previous paragraphs appears to justify this. It's not even clear what alternate scenario you are comparing this to in order to judge that the second star has "no effect" on the kinetic energy of P--for example, are you comparing it with a scenario where there is no Star East, and Star West is given an equal-sized initial push towards P? But in this scenario P might hit Star West less quickly (as measured in terms of P's own proper time) because Star West isn't being pulled in that direction by Star East.

 

[nutgeb]

How are you defining "kinetic energy"? In GR we can use arbitrary global coordinate systems in curved spacetime, and kinetic energy is normally coordinate-dependent. I suppose you could talk about the kinetic energy in the locally inertial frame of an observer on the surface of a star at the moment P reaches that observer,...

Yes, I mean kinetic energy as locally measured by the observer on Star West. He doesn't have to wait for P to arrive, he can observe its approach through a telescope and calculate P's increasing kinetic energy all along its worldline.

It's not even clear what alternate scenario you are comparing this to in order to judge that the second star has "no effect" on the kinetic energy of P--for example, are you comparing it with a scenario where there is no Star East, and Star West is given an equal-sized initial push towards P?

Yes.

But in this scenario P might hit Star West less quickly (as measured in terms of P's own proper time) because Star West isn't being pulled in that direction by Star East.

Well I did some more calculations, which partially changed my view of how this works. As you suggested (or possibly opposite to what you suggested), adding Star East to the mix actually causes P to accelerate toward Star West at a lower rate (and gain less kinetic energy in Star West's frame) than without Star East. But, even with Star East in the mix, as P approaches Star West it does progressively gain kinetic energy in Star West's frame.

Regarding the photon scenario: Over its entire worldline, the photon is continuously climbing out of Star East's gravity well (which causes redshift) and falling into Star West's well (which causes blueshift). But because the stars are farther apart at the start than at the end, the photon falls into Star West's well from a greater initial distance than the eventual distance it climbs out of Star East's well. In other words, the photon started out at higher altitude in Star West's well than the highest altitude it will ever attain in Star East's well. The photon falls farther than it climbs.

This suggests that the photon does experience a net gravitational blueshift in Star West's frame, despite the fact that no gravitational time dilation occurs as between the two stars themselves.

 

[DrGreg]

(For this discussion, we will divide out the component of SR Doppler blueshift so as to ignore it, and focus only on the component of gravitational blueshift).

I'm no expert in this, but I think this might be part of the problem. Decomposing an observed blueshift into "gravitational" and "SR doppler" components is a frame-dependent operation. For example, in the case of a Born-rigid accelerating rocket in flat spacetime, an observer in the rocket attributes the shift between top & bottom as 100% gravitational whereas an inertial observer attributes it as 100% SR doppler.

 

[nutgeb]

I'm no expert in this, but I think this might be part of the problem. Decomposing an observed blueshift into "gravitational" and "SR doppler" components is a frame-dependent operation. For example, in the case of a Born-rigid accelerating rocket in flat spacetime, an observer in the rocket attributes the shift between top & bottom as 100% gravitational whereas an inertial observer attributes it as 100% SR doppler.

Thanks DrGreg, that's an interesting possibility. The observer can't decompose the blueshift if he doesn't know how to attribute it. I usually think of that happening in a "closed" laboratory or spaceship. But here, it seems like the relevant attributes are fully visible to the observer, both the intrinsic gravitational force of Star West and Star East, and the location, velocity and acceleration of both P and Star East relative to the observer. I can't identify any individual component attribute that's hidden, but maybe you can?

 

[nutgeb]

I'm thinking that the concept of gravitational blueshift without any gravitational time dilation might be a component of the cosmological redshift. If it's a real effect, that is.

The asymmetry between a photon's the fall into the observer's gravitational well and its climb out of the emitter's well is much greater in a high-z cosmological model than in the simple two-star model above. At the time a photon left the emitter at, say, z = 1023, the gravitational acceleration acting on the photon from the sphere of cosmic matter centered on the observer was z + 1 = 1024 times greater than the gravitational acceleration from the sphere of matter centered on the emitter is now, just as the photon arrives here. Gravitational acceleration being GM/R.

(Both of these gravity wells actually contain exactly the same total amount of mass-energy, but the well of the emitter now has a (z + 1) times greater radius than the observer's well did then, which dilutes its gravitational potential accordingly.)

Arguably there is no time dilation in the cosmological scenario, because of the symmetry of the cosmological matter distribution around both the emitter and observer. At least that's the case within each epoch (but not across different epochs).

I think it is necessary to find a cosmological solution for a blueshift component that involves no time dilation, because as I've said often before, the FRW metric does not permit time dilation as between comovers.