Physical definition of inertial reference frames

[Goldbeetle]

Dear all,
I'm trying to understand better why gravity makes impossible to physically define an inertial reference frame.

Firstly, we must have an operational procedure that allows us to physically define an inertial reference frame. Secondly, we must show that gravity makes this procedure fail.

What could this operational procedure be? (first question)

I already understand that gravitational red shift desynchronizes clocks after they've been synchronized, whatever the operation procedure.

But what about the space part of the inertial frame definition? I understand by reading a book that this is related somehow to the deflection of light rays caused by gravitation, because according to the book all operational procedures rely on the straight path of light rays. However, the book just mention this without giving any details at all. I understand how deflection happens (it's a consequence of the equivalence principle). But what is the impact on the operational definition of an inertial frame (second question).

I know that locally in space and time we can have an inertial system, but I wanted to understand precisely how come that gravitation destroys any possibility for a physical definition of an inertial frame of reference.

Thanks.
Goldbeetle

PS: the book is "The Riddle of Gravitation" by Bergmann

 

[A.T.]

It is not gravity itself, but the non-uniformity of gravity, that disallows inertial frames of some spatial extend. Free falling objects must have zero coordinate acceleration in an inertial frame. In a non-uniform field there is no such frame that will satisfy this condition globally.

 

[Goldbeetle]

Thanks. But why? This is the sense of my question.

 

[atyy]

An inertial frame is a system of rigid rulers and clocks that don't interact with your experiment in a box.

Gravity is a long range attraction between all stuff. So your experiment is going to bend your rigid rulers.

A concrete example is that general relativity can be written (assuming curved spacetime can be covered by harmonic coordinates) as a field on flat spacetime - ie. global inertial frames exist in general relativity. But the global inertial frame is not directly observable because all your rulers are bent and clocks are slowed. http://arxiv.org/abs/astro-ph/0006423

 

[Goldbeetle]

Atyy,
so bending of rigid rules means distortion of the spatial part of the metric?
What about the operational procedure for inertial frames??
Thanks,
LB

 

[PAllen]

See if this helps. In a global ineratial frame you expect that two stationary objects remain stationary if no force acts on them. Two simple ways this if violated for a non-inertial frame are:

1) uniformly accelerted frame: both objects will accelerated.
2) rotating frame centered on one object: the other will accelerate (in circular motion).

In the case of an attempted large scale inertial frame near a strong gravitating source, if you 'center' the frame on one of the objects, the other accelerates. If you center it elsewhere, they both accelerate. Thus there is no way to meet this simple operational definition of an extended intertial frame.

[Edit: I should state the objects used are light enough so the observed accelerations are orders of magnitude larger than any gravitational interaction between the test bodies]

 

[atyy]

Atyy,
so bending of rigid rules means distortion of the spatial part of the metric?
What about the operational procedure for inertial frames??
Thanks,
LB

Operationally, Lorentz inertial frames are set up by light. From the speed of light we define length (wavelength) and time (frequency). Gravity bends light and red shifts it. So although the global inertial frames still exist (for the most part), if we define them using light, then we don't see the frames.

 

[PAllen]

Operationally, Lorentz inertial frames are set up by light. From the speed of light we define length (wavelength) and time (frequency). Gravity bends light and red shifts it. So although the global inertial frames still exist (for the most part), if we define them using light, then we don't see the frames.

Well, I don't think you see them any other way either (light isn't the only measurement affected), using measurements that work in flat spacetime. At minimum, you derive them as a re-interpretation of measurements.

 

[atyy]

Well, I don't think you see them any other way either (light isn't the only measurement affected), using measurements that work in flat spacetime. At minimum, you derive them as a re-interpretation of measurements.

Yes, that's what I meant. BTW, is that any different from determining Galilean inertial frames?

 

[pervect]

Consider the curved surface of the Earth. If you go x miles north, and y miles east, you wind up in a different spot than if you go x miles east, and y miles north.

This doesn't happen on a flat piece of paper.

So, the thing to look for as evidence of curvature is that it matters as to where you wind up in the above situation.

There are a couple of pieces of mathematical machinery you need to make this work. The first is a piece of machinery that , among other things, let you decide when a line is straight. This is known as "parallel transport". If you start out in some direction, the operation of parallel transport tells you what direction to continue walking in order to walk a straight line.

Parallel transport also tells you, when you've gone your x miles east, what direction is north now.

You can think of this as defining the notion of "parallel lines", which is where the name came from. So, you know what direction is "north" at your starting point, and you know how to define the line parallel to it at your destination, so you know what line is "north" at your destination.

If you happen to have some notion of how to measure distance already, you can define parallel transport from said defintion distance - by a process called "Schild's ladder".

Schild's ladder says basically that if you have a small parallelogram, and the opposing sides have equal length, said parallelogram defines the operation of parallel transport.

You'll need the distance measuring part to decide when you've gone "x miles" and "y miles".

For distance, you can use more or less the usual notion of "the shortest path between two points", replacing the notion of a "straight" line with a geodesic as being the shortest path. You'll notice that this needs a bit of fiddling when you run into situations where there are multiple geodesic paths between two points. But you don't need any fiddling when you consider distance as being something that you define only for "nearby" points, and if you use parallel transport to define the notion of a longer "straight" line, and measure the length of any particular path you want by breaking it up into small pieces and using the method of measuring distance over short intervals that you have.

And that's it, in a nutshell. The notion of "parallel transport" is probably the most abstract notion, "Schild's ladder" is one of the easiest ways to understand it "physically", but I haven't gone into a great amount of detail here, it would probalby be premature at this point.

 

[PAllen]

Yes, that's what I meant. BTW, is that any different from determining Galilean inertial frames?

Only in complexity of re-interpretation, which not a fundamental distinction.

 

[Goldbeetle]

I'm confused.

 

[Mentz114]

I'm confused.

I know that locally in space and time we can have an inertial system, but I wanted to understand precisely how come that gravitation destroys any possibility for a physical definition of an inertial frame of reference.

What do you mean by a 'physical' definition ? What other kind is there ?

 

[Goldbeetle]

For physical definition I mean a procedure by means of which spacetime is coordinates.

 

[Mentz114]

For physical definition I mean a procedure by means of which spacetime is coordinates.

Inertial frames are carried along geodesics. At a point on the 4-D curve we can always find an orthonormal frame, which if it's not spinning will be inertial. Two procedures are ( for instance) Riemann normal coordinates and Fermi normal coordinates. Axes can be constructed from mutually orthogonal null geodesics through the point on the geodesic. Obviously these axes will only be straight in the Euclidean sense over a region. Ultimately this prevents the definition of a global orthonormal frame.