General relativity- Coordinate/metric transformations

In summary, the homework statement asks for a coordinate system (t,x), such that ds2=dt2-dx2 and ds2=dv2-v2du2. The student attempted to solve for the coordinate transformation equations, but got stuck. The equations are easier to solve if they are written in the "comma" notation.
  • #1
jgarrel
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0

Homework Statement


Consider the metric ds2=(u2-v2)(du2 -dv2). I have to find a coordinate system (t,x), such that ds2=dt2-dx2. The same for the metric: ds2=dv2-v2du2.

Homework Equations


General coordinate transformation, ds2=gabdxadxb

The Attempt at a Solution


I started with a general transformation xa→x'a so the new metric is g'μν=gab(dxa/dx'μ)(dxb/dx'ν). The components of g (old metric) and g'(new metric) are known and the unknowns are the derivatives of the old coordinates with respect to the new ones. That's where I'm stuck.
 
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  • #2
jgarrel said:
I started with a general transformation xa→x'a so the new metric is g'μν=gab(dxa/dx'μ)(dxb/dx'ν).
Those derivatives should be partial derivatives. Write it in latex like this: $$g'_{\mu\nu} ~=~ g_{ab}\, \frac{\partial x^a}{\partial x'^\mu} \, \frac{\partial x^b}{\partial x'^\nu} ~.$$ (Look under the Info->Help/How-To menu to find a Latex primer. You'll need to master at least basic latex on this forum.)

Both metrics are diagonal, which makes it reasonably easy to write out the above transformation equations more explicitly.

I.e., write it out explicitly (where ##\mu,\nu## are ##t,x## and ##a,b## are ##u,v##). You should get 2 partial-differential equations where each right hand side has 2 terms. I'll wait to see if you can get that far before giving more hints.
 
  • #3
Thanks for the reply!
I already had ended up with these two differential equations, but I thought there was another way, because they seem difficult to solve. I put them here:

##1=(u^2-v^2) \frac {\partial^2 u} {\partial t^2} -(u^2-v^2) \frac {\partial^2 v} {\partial t^2}##

##-1=(u^2-v^2) \frac {\partial^2 u} {\partial x^2} -(u^2-v^2) \frac {\partial^2 v} {\partial x^2}##
 
  • #4
Try writing out the corresponding PDEs for the inverse transformation first. I.e., treating ##t,x## as functions of ##u,v##. They turn out a bit easier.

Btw, if those partial derivatives become too tedious to keep writing out fully in latex, you can always use the briefer "comma" notation, e.g., $$ \frac{\partial u}{\partial t} ~\equiv~ u,_t ~~;~~~~ \mbox{and}~~~~~~ \frac{\partial^2 u}{\partial t^2} ~\equiv~ u,_{tt} ~~;~~~~ \mbox{(etc)}.$$
 
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Related to General relativity- Coordinate/metric transformations

1. What is general relativity?

General relativity is a theory of gravity developed by Albert Einstein in 1915. It describes how the force of gravity arises from the curvature of space and time caused by the presence of massive objects.

2. What are coordinate transformations in general relativity?

Coordinate transformations refer to changes in the way we measure space and time in different reference frames. In general relativity, these transformations are used to describe how the geometry of space-time changes in the presence of massive objects.

3. How are metric transformations related to general relativity?

Metric transformations are used in general relativity to describe the relationship between space-time coordinates and the curvature of space-time caused by the presence of massive objects. The metric tensor is a mathematical object that represents this relationship.

4. Why are metric transformations important in general relativity?

Metric transformations are important in general relativity because they allow us to describe how space-time is curved in the presence of massive objects. This is essential for understanding the effects of gravity on the motion of objects and the behavior of light.

5. How do coordinate and metric transformations affect our understanding of gravity?

Coordinate and metric transformations in general relativity change our understanding of gravity by showing that it is not a force between objects, but rather a result of the curvature of space-time caused by massive objects. This explains why objects with mass are attracted to each other without the need for an external force.

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