Inertial reference frames in general relativity

[analyst5]

Hi guys,

I was reading some stuff about general relativity and the first impression is that isn't completely the follow-up to SR. In what I've read it has a different assumption about inertial frames, that they are only significant only locally. What does this really mean and what are the differences in the conception of space time between GR and SR in this manner?
Can all bodies that are in a uniform state of motion be considered inertial reference frames, but the only difference is that their plane of simultaneity has a local (limited spacetime area) meaning instead of global?

I appreciate your answers and opinion.
Regards.

 

[WannabeNewton]

A freely falling material particle is locally inertial in the sense that at any given point, we can find a coordinate system in which the particle is traveling on a straight line, in the Euclidean sense, at that point. In SR on the other hand we can construct global inertial coordinate systems.

 

[kevinferreira]

It is called the equivalence principle which, together with the relativity principle, make up the whole theory. The equivalence principle says

A little reflection will show that the law of the equality of the inertial and gravitational mass is equivalent to the assertion that the acceleration imparted to a body by a gravitational field is independent of the nature of the body. For Newton's equation of motion in a gravitational field, written out in full, it is:

(Inertial mass) (Acceleration) = (Intensity of the gravitational field) (Gravitational mass).

It is only when there is numerical equality between the inertial and gravitational mass that the acceleration is independent of the nature of the body.

as Einstein himself wrote.
This allowed him to conceive the idea that maybe gravitation was indistinguishable from an inertial effect, when seen locally. This only holds locally because of other effects, as tidal forces, consequence of the nature of the gravitational field.
The identification of gravity as an inertial effect when seen locally (equivalence principle) allows one to 'put aside' gravitational effects simply by chosing the right frame, as in usual Newtonian mechanics. But only locally. This is a justification for the mathematical statement of spacetime as a manifold, which is an object that is locally flat (euclidian), but can be whatever non locally.

Moreover, the equivalence principle is pushed a bit more far and leads Einstein to think that maybe gravitation is purely kind of an inertial effect, simply by thinking of test particles under gravitational influence to be free particles in curved spacetimes. This gives a nice way of 'completing' the whole reasoning.

 

[analyst5]

A freely falling material particle is locally inertial in the sense that at any given point, we can find a coordinate system in which the particle is traveling on a straight line, in the Euclidean sense, at that point. In SR on the other hand we can construct global inertial coordinate systems.

Hey WannabeNewton,
are there any significant changes that are not connected to points/objects that are not free falling. So does the hyperplane of simultaneity narrow down in space-time, as I've red in some papers... or do intertial frames exist at all (which is quite dramatic to say) in GR?

Thank you

 

[WannabeNewton]

Hi analyst5! In GR we have two different concepts: that of a local hyperplane of simultaneity, which you alluded to in your first post, and that of a global hyperplane of simultaneity which involves a time slice defined by a family of observers situated throughout space-time. To that end, see the following two links and the further references given in the links (if you can get access to them):

http://www.socsci.uci.edu/~dmalamen/bio/GR.pdf (pages 162-164)
http://arxiv.org/pdf/gr-qc/0506127v1.pdf

If you can get access to Rindler's paper (referenced in the first link) then that would certainly be a good read; I tried looking online to see if there's a public access version of it but I couldn't fine one. This springer link is all I could find: http://link.springer.com/content/pdf/10.1007/BF00756593.pdf. The following paper might also be of use if you can get access to it as well: http://jmp.aip.org/resource/1/jmapaq/v21/i7/p1783_s1?isAuthorized=no . I can access both of them for free using my Cornell university account so if you have a university account then use that to get access. Cheers.

EDIT:

...or do intertial frames exist at all (which is quite dramatic to say) in GR?

I missed this part before but as you already know inertial frames exist only locally.

 

[analyst5]

Hi analyst5! In GR we have two different concepts: that of a local hyperplane of simultaneity, which you alluded to in your first post, and that of a global hyperplane of simultaneity which involves a time slice defined by a family of observers situated throughout space-time. To that end, see the following two links and the further references given in the links (if you can get access to them):

http://www.socsci.uci.edu/~dmalamen/bio/GR.pdf (pages 162-164)
http://arxiv.org/pdf/gr-qc/0506127v1.pdf

If you can get access to Rindler's paper (referenced in the first link) then that would certainly be a good read; I tried looking online to see if there's a public access version of it but I couldn't fine one. This springer link is all I could find: http://link.springer.com/content/pdf/10.1007/BF00756593.pdf. The following paper might also be of use if you can get access to it as well: http://jmp.aip.org/resource/1/jmapaq/v21/i7/p1783_s1?isAuthorized=no . I can access both of them for free using my Cornell university account so if you have a university account then use that to get access. Cheers.

Thank you very much on the links. I'll read them when I come back home from work.

 

[analyst5]

I've red the link you gave me and it is certainly very educative, I just need time to make the puzzles come together in my head.

Another idea about Irf-s came to my mind, how does rotation affect them. For instance we know that the Earth is rotating around the sun, it has some form of proper rotation, therefore the bodies on it rotate around the sun too. Do they lose their status of inertial reference frames because of rotation, or is the idea that they rotate around something compatible with their status as inertial reference frames(of course if they meet other criteria, like constancy of motion, rectlinearity etc.)

Thank yoz

 

[Nugatory]

Do they lose their status of inertial reference frames because of rotation?

Yes.
Of course we can ignore the rotation if the effects are small enough. For example,when I drop a ball from my hand to the floor the Coriolis effect caused by the rotation of the Earth affects the ball's motion very slightly so that it does not follow the exact trajectory we'd expect in an inertial frame. However, this effect is so tiny that we ignore it (indeed, it's very hard to detect even if you're looking for it with extraordinarily sensitive instruments), do the math as if we're in an inertial frame.

But on the other hand if we're considering a continent-sized mass of warm air moving across an ocean then the non-inertial effects from the Earth's rotation are very visible and cannot be ignored - they can set the air mass spinning and turn it into a cyclone.

 

[analyst5]

Yes.
Of course we can ignore the rotation if the effects are small enough. For example,when I drop a ball from my hand to the floor the Coriolis effect caused by the rotation of the Earth affects the ball's motion very slightly so that it does not follow the exact trajectory we'd expect in an inertial frame. However, this effect is so tiny that we ignore it (indeed, it's very hard to detect even if you're looking for it with extraordinarily sensitive instruments), do the math as if we're in an inertial frame.

But on the other hand if we're considering a continent-sized mass of warm air moving across an ocean then the non-inertial effects from the Earth's rotation are very visible and cannot be ignored - they can set the air mass spinning and turn it into a cyclone.

But doesn't SR lose some credibility then? I don't of course mean it's false, but is it used in defining temporal order of events on Earth and nearby space, relative to some frame of reference? I mean if inertial frames are the basis of relativity it seems weird that rotation disallows them, and disallows various judgements of simultaneity that sR presupposes.

 

[Dale]

I don't know what you mean by credibility in this context.

You can use SR any time tidal gravity is negligible. SR can handle rotation and acceleration just fine. They are non inertial, but the math is well understood.

 

[analyst5]

I don't know what you mean by credibility in this context.

You can use SR any time tidal gravity is negligible. SR can handle rotation and acceleration just fine. They are non inertial, but the math is well understood.

So is it valid to use some body on the Earth as a reference frame for judgement of simultaneity, velocities etc. I didn't meant credibility in a sense of its truth, but rather can it be used in a practical context despite rotation, like in my previous sentence. And thanks both to you and Nugatory for the replies.

 

[Nugatory]

But doesn't SR lose some credibility then? I don't of course mean it's false, but is it used in defining temporal order of events on Earth and nearby space, relative to some frame of reference? I mean if inertial frames are the basis of relativity it seems weird that rotation disallows them, and disallows various judgements of simultaneity that SR presupposes.

Inertial frames are not the basis of special relativity; SR works in any flat spacetime, whether the frames are inertial or not. It's just that most introductory texts use inertial frames because the math is simpler, doesn't get in the way of explaining the concepts; and then people who don't study beyond the introductory texts don't get to see SR at work in a non-inertial frame.

If you google around for "Rindler coordinates" you'll find one particularly interesting but still fairly simple example of SR in a non-inertial frame.

 

[Dale]

So is it valid to use some body on the Earth as a reference frame for judgement of simultaneity, velocities etc.

Certainly, but if the object is non-inertial then the reference frame will be non-inertial also. There is nothing wrong with that, but most of the famous formulas are simplified for inertial frames so you have to use the more general but less famous equations.

 

[analyst5]

Certainly, but if the object is non-inertial then the reference frame will be non-inertial also. There is nothing wrong with that, but most of the famous formulas are simplified for inertial frames so you have to use the more general but less famous equations.

So, nonetheless, objects on Earth can have different simultaneity planes of events on earth, time dilation and length contraction also occur. So 'SR handles rotation' means that the objects that rotate are affected by the same relativistic effects as inertial reference frames? Objects on Earth also become length contracted in a frame of another object, that is let's say also moving relative to the earth, for an example. Is this a clear enough distinction?