0
$\begingroup$

Hoping my English would not be confusing.Glad to hear some language suggestion🤣

I have seen many posts in stacks,for example,

How to understand the true meaning of momentarily comoving reference frame?

Locally flat coordinate and Locally inertial frame

But they seem a bit different from my question.

First,if we introduce the manifold theory at the beginning of GR,then I think the locally inertial reference frame can be described by "If our spacetime is a metric manifold,then its metric tensor is locally Minkowskian"I mean if we find a coordinate system at $p$ such that the metric tensor can be written as Minkowskian metric,we will say the coordinate system is a locally inertial reference frame at $p$.

I guess my understanding is correct,but there is a subtle thing that I can't really understand.

LIRF for locally inertial reference frame.And EP for equivalence principle

According to equivalence principle,physical phenomenon in the LIRF is identical to the phenomenon in $M^4$,so I think the LIRF is moving,and it can be thought as one certain coordinate system!

To be more precise,Let's consider a persen A free falling in gravitational field,and a person B in $M^4$,according to EP,if B choose the coordinate system that B is always at the origin point,and do some experiment (in a small region),then A can choose the coordinate system that A is always at the origin point and get the same result.

Alright,now I encounter a problem,A will choose a "moving coordiante",which is a series of coordinate system,and not "a" coordinate system in which the metric tensor can be written as $\eta$.And this seems also cause a mathematical complexity,because we will use many different charts to describe one phenomenon (like a curve in the manifold)

I am not sure whether I confuse the two different definition "comoving reference frame" and LIRF.

I also studied a little(alright,2 hours in total) Cartan's moving frame.But I can't see its direct connection to GR,but its said the moving frame sometimes used to define reference frame,how?

A long question...Thank you for your answering.

Improve this question
$\endgroup$
6
  • $\begingroup$ A comoving reference frame is with respect to something else, say a body. It is a reference frame which moves along with that body and it needs not be inertial $\endgroup$
    – Damian
    Commented Jul 24 at 16:01
  • $\begingroup$ What is M4 and what does it mean for B to be in M4? $\endgroup$
    – Dale
    Commented Jul 24 at 16:16
  • $\begingroup$ @Dale $M4$ for the 4 dimensional Minkowskian manifold,physically it would be an empty universe I guess $\endgroup$
    – MakiseKurisu
    Commented Jul 25 at 2:07
  • $\begingroup$ Ok, so in the context of the problem B is a theoretical observer in a theoretical flat spacetime and A is a real observer in free fall in the real universe curved spacetime? $\endgroup$
    – Dale
    Commented Jul 25 at 2:27
  • $\begingroup$ @Damian I see.Thanks.So mathematically is it a series of charts rather than a fixed chart? $\endgroup$
    – MakiseKurisu
    Commented Jul 25 at 2:42

1 Answer 1

Reset to default
1
$\begingroup$

A will choose a "moving coordiante",which is a series of coordinate system,and not "a" coordinate system in which the metric tensor can be written as η.

This is not correct. Time is part of the spacetime manifold and coordinate charts include time and cover time. So A’s coordinate system is indeed a single coherent coordinate chart. This coordinate system is known as a Riemannian normal coordinate system. The existence of such charts locally along every geodesic is a basic theorem of differential geometry.

Improve this answer
$\endgroup$
4
  • $\begingroup$ Thanks.But I still have some questions... $\endgroup$
    – MakiseKurisu
    Commented Jul 25 at 2:47
  • $\begingroup$ Oh I see! By the way is the Riemann normal coordinate induced by the exponential map.let $v$ be a vector in tangent space,then we set the coordinate at exp($v$) as the coordinate of $v$ in the tangent space? My math department friend mentioned a term Gauss normal coordinate defined as above,if they are the same I can ask him for more information. Thanks again!🤩 $\endgroup$
    – MakiseKurisu
    Commented Jul 25 at 3:05
  • $\begingroup$ @MakiseKurisu yes, it uses the exponential map, but I don’t know about Gauss normal coordinate. $\endgroup$
    – Dale
    Commented Jul 25 at 3:18
  • 1
    $\begingroup$ Seems they're just different term for the same thing.But "Riemann coordinate" is more frequent.And finally I find what I want in physics is the Fermi coordinate😆 $\endgroup$
    – MakiseKurisu
    Commented Jul 26 at 15:12

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.