Need help understanding frames of reference in GR

  • #1
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Hi I have recently started GR and have found the mathematics to be quite easy (have encountered differential manifolds and tensor calculus in other subjects), but the physics is troubling me, allow me to elaborate.
In special relativity, we have a very intuitive idea of how observations work, there are global inertial frames as in classical mechanics, and to link one inertial observer to another, we simply use the Lorentz transform. I find this very intuitive, even though the concepts of absolute time and distance have been abandoned, the invariant interval makes up for that.
However, I have no idea how things are suppose to work in GR.
I understand that the equality between inertial mass and gravitational mass rules out any possibility for a global inertial frame, but that's where my intuition drops dead.
How are observers treated in GR?
 

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  • #2
Bill_K
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In special relativity, reference frames are global. In general relativity, reference frames are local. An observer is represented by a world line, and he directly observes only events on that line. Any information from events not on his world line is received via light signals.
 
  • #3
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I guess I should make my question more specific, I do not really understand what it means physically to construct a local Lorentz frame, are these frames simply mathematical or do they have a physical meaning ? (Like free falling frames, which have the metric and it's first derivatives vanish along it right? (By "along", I mean along the world line of the free falling frame))
 
  • #4
pervect
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I guess I should make my question more specific, I do not really understand what it means physically to construct a local Lorentz frame, are these frames simply mathematical or do they have a physical meaning ? (Like free falling frames, which have the metric and it's first derivatives vanish along it right? (By "along", I mean along the world line of the free falling frame))
The metric is diagonal in a local Lorentz frame. Furthermore in geometric units the absolute value of the metric is a unit diagonal matrix, i.e the metric is either diag(-1,1,1,1) or diag(1,-1,-1,-1), depending on your choice of conventions. No statement is made about the derivatives of the metric, though. The 3+1 signature is required to make the space-time Lorentzian, i.e. you can have one minus sign, or three, but not 0,2, or 4.

[add]This is using the defintions of Local Lorentz frames from MTW"s Gravitation, quoted from memory.
 
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  • #5
pervect
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(Like free falling frames, which have the metric and it's first derivatives vanish along it right? )
Oh, I wanted to ask where this definition of "free falling frames" came from, while I was at it. Is this published somewhere? If it is , I'd like to track down and read the reference if possible.
 
  • #6
zonde
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I understand that the equality between inertial mass and gravitational mass rules out any possibility for a global inertial frame, but that's where my intuition drops dead.
How are observers treated in GR?
Do you mean - what observers see in GR and what system they can use to arrange their observations?
Or something more discrete like in post #3?
 
  • #7
pervect
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The easiest solution for observers in GR is "don't need them and don't want them", but I'd guess it's not the only solution. It's probably the easiest and least ambiguous one though,.

For a paper on this approach see http://arxiv.org/abs/gr-qc/9508043 "Precis on General Relativity" by Misner.

One first banishes the idea of an “observer”. This idea aided Einstein
in building special relativity but it is confusing and ambiguous in general
relativity. Instead one divides the theoretical landscape into two categories.
One category is the mathematical/conceptual model of whatever is happening
that merits our attention.

The other category is measuring instruments and the data tables they provide.
More on the conceptual model follows

What is the conceptual model? It is built from Einstein’s General Relativity
which asserts that spacetime is curved. This means that there is no
precise intuitive significance for time and position. [Think of a Caesarian
general hoping to locate an outpost. Would he understand that 600 miles
North of Rome and 600 miles West could be a different spot depending on
whether one measured North before West or visa versa?] But one can draw
a spacetime map and give unambiguous interpretations. [On a Mercator
projection of the Earth, one minute of latitude is one nautical mile everywhere,
but the distance between minute tics varies over the map and must
be taken into account when reading off both NS and EW distances.] There
is no single best way to draw the spacetime map, but unambiguous choices
can be made and communicated, as with the Mercator choice for describing
the Earth.

The conceptual model for a relativistic system is a spacetime map or
diagram plus some rules for its interpretation. For GPS the attached Figure
is a simplified version of the map. The real spacetime map is a computer
program that assigns map locations xyzt to a variety of events.
To make it short and sweet, the "conceptual model" is the metric. This is the "space time map", the one that gives labels to events (t,x,y,z) and allows you to calculate the Lorentz interval between events (and also distances and times).

Anything that an "observer" on a specified wordline could observer can be determined from the metric - readings of clocks, transmission and reception of radar signals, etc.

As I mentioned on another thread, Fermi Normal Coordinates can be a great intuitive aid and serve as a replacement for the tradional "frame of reference", because they are a special set of coordinates that make everything look as close to Newtonian as possible.

So if you are confused about , for instance, what notion of simultaneity to use, using the Fermi normal notion of simultaneity is convenient because it makes the dyamics of the system look very nearly Newtonian, which is what one is used to.

If you don't care about making things look as nearly Newtonian as possible, and are wiling to do the calculations, you can use any coordinates you like, of course. And any notion of simultaneity, as well.

The only bad thing about Fermi Normal coordinates is it may be difficult to learn them when you are struggling with the very basics of understanding GR. So on the overall I'd recommend "banishing the observer" first, especially if the papers on Fermi Normal coordinates seem too technical. You don't really need "an observer", you really only need to know how to work with the metric.
 

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