Applicability of GR in Tidal Gravity Problems?

[metastable]

Is General Relativity applicable in problems involving tidal gravity? For example if a system being analyzed is large enough where tidal gravity effects become apparent - suppose two distantly separated hovering observers see a freefalling object change velocity - are general relativity equations appropriate?

The reason I ask is I was reading the following article:

https://en.m.wikipedia.org/wiki/Introduction_to_general_relativity#Tidal_effects

and it states:

“The equivalence between inertia and gravity cannot explain tidal effects – it cannot explain variations in the gravitational field.[10] For that, a theory is needed which describes the way that matter (such as the large mass of the Earth) affects the inertial environment around it.”

 

[PeterDonis]

Is General Relativity applicable in problems involving tidal gravity?

Every problem in which GR is applicable will have tidal gravity, since tidal gravity is spacetime curvature, and spacetime curvature defines the problems in which GR is applicable (if spacetime is flat, SR is sufficient).

The equivalence between inertia and gravity cannot explain tidal effects

Yes, and that is because it can be present in flat spacetime, with no actual gravitating masses anywhere. Just get in your rocket and fire the engine, then hold an object and release it. It will accelerate towards the floor of the rocket, just as if you were standing on the surface of a planet. In other words, "gravity" is present in the rocket. But tidal gravity is not, because spacetime is flat.

For that, a theory is needed which describes the way that matter (such as the large mass of the Earth) affects the inertial environment around it.

And that theory is General Relativity.

 

[Dale]

Is General Relativity applicable in problems involving tidal gravity?

Tidal gravity = curved spacetime. So yes, GR is applicable to tidal gravity.

You misunderstand the use of the equivalence principle here. The equivalence principle is used to avoid GR and just use SR in problems without tidal gravity. For problems with appreciable tidal gravity you cannot use SR only but must use GR instead.

 

[metastable]

And that theory is General Relativity.

So yes, GR is applicable to tidal gravity.

So if I understand correctly what everyone is saying, GR equations correctly won't imply a 0.999999c upward electron (to hovering observer) will decelerate at roughly ~1m/s in the nearly uniform 1m/s^2 region around TON 618.

 

[Dale]

I am not sure what TON 618 is, but if the gravitational field is nearly uniform then to a good approximation you can just use SR.

 

[metastable]

"TON 618 is a very distant and extremely luminous quasar—technically, a hyperluminous, broad-absorption line, radio-loud quasar—located near the North Galactic Pole in the constellationCanes Venatici. It contains one of the most massive known black holes, weighing 66 billion times the mass of the Sun.[2]"
https://en.wikipedia.org/wiki/TON_618

 

[PAllen]

"TON 618 is a very distant and extremely luminous quasar—technically, a hyperluminous, broad-absorption line, radio-loud quasar—located near the North Galactic Pole in the constellationCanes Venatici. It contains one of the most massive known black holes, weighing 66 billion times the mass of the Sun.[2]"
https://en.wikipedia.org/wiki/TON_618

The bigger the BH, the less tidal gravity there is in the vicinity of a given proper acceleration for a stationary observer, thus the better the approximation of using SR (the equivalence principle) is. Thus, your choice of this body suggests the opposite of what you intended. For example, near earth, the deviation from the principle of equivalence is about 1 part in a billion per meter. For the region near this BH where the proper acceleration of a stationary observer is 1 g, the deviation from the POE would be many orders of magnitude smaller than this.

 

[metastable]

Well I'm very confused now -- in the thread that was closed (about particle deceleration in relativistic jets) I was told that GR equations are the only ones to use to calculate the deceleration of electrons in relativistic jets.

 

[PeterDonis]

if I understand correctly what everyone is saying, GR equations correctly won't imply a 0.999999c upward electron (to hovering observer) will decelerate at roughly ~1m/s in the nearly uniform 1m/s^2 region around TON 618.

If you are asking a question because of something discussed in a previous thread, you need to say so explicitly. Nobody up until this post of yours had any idea that you were asking about your particular scenario in that previous thread. Your question in the OP of this thread is just a general question about GR.

I am not sure what TON 618 is

He's referring to this previous thread, which was closed:

https://www.physicsforums.com/threads/particle-deceleration-in-relativistic-jets.973635/

 

[metastable]

@PeterDonis I was looking into the GR equations, as you suggested Mr. Donis, but I wanted to make sure they weren't going to give the wrong answer before I spend too much time on them.

 

[PeterDonis]

I'm very confused now

That's because you failed to provide important information to the other posters in this thread--the fact that your question was prompted by a discussion in a previous thread and was not just a general question about GR. The discussion in that previous thread provides crucial context that the other posters did not have. You can't expect to get useful answers if you don't give others all the relevant information.

in the thread that was closed (about particle deceleration in relativistic jets) I was told that GR equations are the only ones to use to calculate the deceleration of electrons in relativistic jets.

And for the particular problem you posed in the OP of that thread, that is true. You were talking about relativistic jets that extend for thousands of light-years from black holes at the centers of quasars. The equivalence principle cannot be applied over those distances; tidal gravity is not negligible and the gravitational field is not uniform.

For the more limited problem you retreated to during the discussion in that previous thread, a gravitational field with uniform acceleration of 1 m/s^2 over a distance of 1 light-second, you could use SR, but not the SR equations you were using. You kept on using the SR equations for objects decelerated by a force. Gravity is not a force in GR, and the electrons being "decelerated" by gravity in the jets are in free fall. You could use the SR equations for a body in free fall moving relative to a family of uniformly accelerating observers (i.e., Rindler observers) for this limited case, which is what @Dale and @PAllen are referring to in their posts earlier in this thread. However, those were not the SR equations you were using in the previous thread, and they won't work for a full analysis of the jets anyway. Whereas, the correct GR equations for a body free-falling in a gravitational field are simple enough even for the limited case and work for a full analysis of the jets.

 

[PeterDonis]

I was looking into the GR equations, as you suggested Mr. Donis, but I wanted to make sure they weren't going to give the wrong answer before I spend too much time one them.

This is just silly. If the GR equations were to give the wrong answer, the SR equations would too, since they are just special cases of the GR equations. You need to stop making excuses and do the work.

 

[Dale]

@PeterDonis I was looking into the GR equations, as you suggested Mr. Donis, but I wanted to make sure they weren't going to give the wrong answer before I spend too much time on them.

The GR equations will give you the right answers. Sometimes you you can get away with using the SR equations as an approximation via the equivalence principle. That is an inaccurate approximation to the accurate GR calculation. I.e. the GR calculations are the right answer, it is other approaches that may get wrong answers depending on the details.

I looked at that thread. With the absurd level of precision you are using to do your calculations you will need GR. If you used more reasonable numbers over a reasonably small distance then you could use the equivalence principle and SR instead. Or better yet use variables and only plug in numbers at the end.

 

[Dale]

The equivalence principle cannot be applied over those distances; tidal gravity is not negligible and the gravitational field is not uniform.

Particularly not at the levels of precision that he was using. The more digits of precision you use the less distance you can go before it makes a difference. He is using an astounding number of significant figures, so tidal effects will become important over very small distances, possibly microscopic distances.

 

[pervect]

The principle of equivalence is often presented as one of the cornerstones of GR, yet it can be very confusing.

I like to stress what I feel is the least confusing version of the equivalence principle, the weak equivalence principle.

The weak equivalence principle just says that under identical conditions, all bodies fall at the same rate. So if we drop two objects under indentical conditions, it doesn't matter what the objects are made of, they both fall at the same rate.

Working out the details of "identical conditions" is the part that can get confusing. And we occasionally see a lot of wrangling about it. The focus, though, is not on what constitues "identical conditions", the foucs is on that it doesn't matter what the body is made of. The fine details of "identical conditions" eventually need to be worked out, but they're not the thrust of the principle.

Note that this version of the equivalence principle applies both for the "fields" of an actual mass, and the behavior of a body on Einstein's elevator. In the later case in particular, it's pretty clear that if you don't apply force on a body, the body will stay still, regardless of what material it's on, at least if the behavior approximates Newton's theory, as it must.

As far as "identical conditions" go, to get truly identical conditions it's probably best to imagine that your'e dropping the two test bodies (of different materials) from the same place. If you drop two bodies from different places, you'll notice very small effects, such as the bodies converging in their trajectories as they fall towards the center of the Earth when they are separated horizally, or bodies arriving at the surface of the Earth at different times if they are dropped vertically from different heights. This does not violate the principle of equivalence in any way, it's just some commentary on what makes conditions exactly identical vs what makes conditions "almost" identical.