Thread: GR question

1. Order of Kilopi
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Nov 2002
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GR question

I have been studying differential geometry, on my own, for about the last year. This was because I wanted to understand GR better. I now feel somewhat comfortable with dg (the is a whole lot that I am still working and struggling with), but I am either having trouble applying it to GR or, if I can apply it, figuring out a description for what is going on in the math. I know I should get either Wald or MTW, but finances, at this time prohibit it. So, to my question. If the Christoffel symbol at an arbritray point on the Manifold is 0, it seems that the curvature at that point goes away. If this is the case, does that mean that that point is an inertial frame? And if this is so, is this the mathematical expression for the equivalence principle?

edited for spelling

2. Re: GR question

Originally Posted by Tensor
[Snip!]If the Christoffel symbol at an arbritray point on the Manifold is 0, it seems that the curvature at that point goes away. If this is the case, does that mean that that point is an inertial frame? And if this is so, is this the mathematical expression for the equivalence principle?
Remember that the curvature involves the derivative of the Christoffel sysmbol and that is not necessarily zero at a point.. Essentially, at any given point you can transform the coordinates in order to make the metric equal to the Minkowski metric at that point and the derivatives of the metric all zero at that point, but in general you cannot make all of the second derivatives of the metric zero at a point.

3. Order of Kilopi
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Re: GR question

Originally Posted by Celestial Mechanic
Originally Posted by Tensor
[Snip!]If the Christoffel symbol at an arbritray point on the Manifold is 0, it seems that the curvature at that point goes away. If this is the case, does that mean that that point is an inertial frame? And if this is so, is this the mathematical expression for the equivalence principle?
Remember that the curvature involves the derivative of the Christoffel sysmbol and that is not necessarily zero at a point..
This would involve the Ricci curvature Tensor, right?

Originally Posted by Celestial Mechanic
Essentially, at any given point you can transform the coordinates in order to make the metric equal to the Minkowski metric at that point
The transforms are one of the parts I am struggling with.

Originally Posted by Celestial Mechanic
and the derivatives of the metric all zero at that point, but in general you cannot make all of the second derivatives of the metric zero at a point.
So what, mathematically, constitutes an inertial frame? And what, mathematically, is the basis for the equivalence principle?

I apologize if these questions sound simplistic (to some of you) and not quite in the right order, but I think I'm skipping around some picking things up as I can find them.

4. Re: GR question

Originally Posted by Tensor
So what, mathematically, constitutes an inertial frame? And what, mathematically, is the basis for the equivalence principle?
I can barely spell GR but I might can help with this one.

Think of all the coordinates fixed around you and your computer. They are not "moving". This is one inertial frame. A car going down a straight road and at a constant speed is another inertial frame. Rectilinear and fixed speed for you or an object so that all the coordinates in it's space are fixed is an inertial frame. If you drop a ball inside your car, traveling at any fixed speed, the ball will fall downward and you, and others not in the car, will measure all motion nicely. You see the ball go straight down with 0 forward motion. If your car is doing 70 then your ball is doing 70 in the eyes of those standing on the ground in their inertial frame of reference.

There are an infinite number of inertial frames, obviously. Newtonian view held to an absolute inertial frame out there. It wasn't then, and isn't now, necessary to know it in order to calculate relative motions as shown above. I wonder if they thought the ether wind was 0 in this frame?

Of course, special relativity changed the meaining of "measuring motion nicely". We don't have an absolute space but we do have an absolute speed.

5. Order of Kilopi
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Re: GR question

Originally Posted by George
Originally Posted by Tensor
So what, mathematically, constitutes an inertial frame? And what, mathematically, is the basis for the equivalence principle?
Think of all the coordinates fixed around you and your computer. They

snip

nicely". We don't have an absolute space but we do have an absolute speed.
George, while I appreciate the explanation, what I was asking was what do you have to do to the GR equations to get the equivalent of an inertial frame? And what, in the GR equations, is the basis for the equivalence priciple. I think that they are tied together, but I'm not sure what it looks like mathematically (in the GR equations).

6. Oops, I feared as much considering the discussions. You did use the term - "simplistic" You caught me just after I bought Max Born's book on relativity and I am going through the foundational thoughts.

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