Constructing Local Lorentz Frames in Curved Spacetime

[tsahi]

hi,

g.r speeks of the ability to constract local lorentz frame. how can an observer
construct such a frame if spacetime is curved? what are his rods and clocks?
it seems that if one tried to construct a "hive" of coords using a ruler,
then it will not cross as expected... he might even notice that
tiangle angles do not sum to 180...
i guess it all depends on accuracy of measurement but i find it hard
to visualize and calculate that accuracy. it also is hard for me to
understand how increased accuracy will cause tidal forces. increased accuracy
means that can't construct flat space and therefore can't measure tidal forces...
please help and elaborate an explanation if you have one...
i am going crazy here (and out of hair).

 

[JesseM]

g.r speeks of the ability to constract local lorentz frame. how can an observer
construct such a frame if spacetime is curved? what are his rods and clocks?

A local Lorentz frame only exists in an infinitesimally small region of spacetime, it's really an idea based on limits. If you have a freefalling observer in a box, then in the limit as the size of the box goes to zero and the time during which he makes his observations also goes to zero, the effects of tidal forces due to curved spacetime will go to zero, and measurements inside this region will become arbitrarily close to those made inside an identical box moving inertially in flat spacetime.

 

[Fredrik]

The local Lorentz frame is usually well-defined in a large region, but it only agrees with a grid of rulers and clocks in an infinitesimal region.

We don't even have to consider GR (i.e. curved spacetime) to run into this problem. We have the same problem with accelerating frames in SR (i.e. in Minkowski space). This is probably why some people consider accelerating frames in Minkowksi space to be a part of GR rather than SR.

 

[tsahi]

does anyone know of a text which derives exactly the error of Newtonian coordinates?
the error of streching coordinates out from their local flatness?
also, when in a small region, MTW (a.k.a the phonebook), speeks of making measurements more precise and then noting tidal effects. i did
not understand that. if we refine measurements we are no more in the local flatness
and tidal effects cannot be analysed Newtonianly (like the book does)...

 

[atyy]

Try Ohanian. He has a describes a device that measures tidal effects. He also had a paper in Physical Review or something many years back about the same thing. I am a bit confused whether tidal forces disappear in an infinitesimal region of spacetime. Many books say that, but Ohanian doesn't. Rindler has statements similar to Ohanian. I believe the correct statement is that in a curved space, a local Lorentz frame can be defined at each point such that the metric is flat up to first order, but not second. Tidal effects and the Riemann tensor are second order effects, and so can be seen even at a point (since the second derivative exists at each point). So perhaps MTW mean going to second order when they talk about making measurements more precise.

 

[atyy]

http://www.pma.caltech.edu/Courses/ph136/yr2006/text.html

In Chapter 24, Blandford and Thorne treat the issue of the local Lorentz frame and first and second order derivatives carefully. As far as I can tell, the essential steps are to establish a local Lorentz frame so that the metric is not flat only at second order and higher. Then take the low velocity approximation so that coordinate and proper time can be identified. Finally, compare the second order derivatives of the metric with the Newtonian tidal forces expression.

 

[tsahi]

How come, the newtinian limit conditions are defined as they are?

 

[atyy]

How come, the newtinian limit conditions are defined as they are?

I guess the idea is that General relativity "incorporates" Special relativity and Newtonian gravity. So to get the Newtonian limit, we have to get rid of special relativity, which we do as usual by taking the low velocity limit.