Coordinate transformation into a standard flat metric

offscene
Messages
7
Reaction score
2
Homework Statement
Starting from the metric ##ds^2=-X^2dT^2+dX^2##, using a coordinate transformation of T and X to t and x, convert it to the form ds^2=-dt^2+dx^2 (standard flat 2D metric).
Relevant Equations
Not actually a homework problem ("confession" from example 7.3 in Hartle's book for Gravity and GR) and no other relevant equations I can think of besides the standard chain rule but thought that this was the most fitting place to ask.
I started by expanding ##dx## and ##dt## using chain rule:

$$dt = \frac{dt}{dX}dX+\frac{dt}{dT}dT$$
$$dx = \frac{dx}{dX}dX+\frac{dx}{dT}dT$$

and then expressing ##ds^2## as such:

$$ds^2 = \left(\left(\frac{dt}{dX}\right)^2+\left(\frac{dt}{dX}\right)^2\right)dX^2+\left(\left(\frac{dt}{dT}\right)^2+\left(\frac{dt}{dT}\right)^2\right)dT^2 + 2\left(\frac{dt}{dX}\frac{dt}{dT}+\frac{dx}{dT}\frac{dx}{dX}\right)$$

But after matching the coefficients to the original ##ds^2##, I am unable to solve the equations to come up with the right transformation and was wondering if anyone could point me in the right direction/show me.
 
Physics news on Phys.org
Nevermind, I found the transformation with ##x = X\cosh(T)## and ##t = X\sinh(T)## with some guess and check but is there a cleaner way to do this?
 
  • Like
Likes vanhees71 and PeroK
The null geodesics in the usual Minkowski coordinates are given by ##x\pm t = const##. You can find them in the given coordinates and use that to set the coordinate change that matches them. In your case the null curves are given by ##dX^2=X^2dT^2##, which can be solved easily and gives ##Xe^{\pm T} = const## (you don't have to check that these are geodesics, if the change of variables works). So setting ##x+t = Xe^T## and ##x-t = Xe^{-T}## gives you the ones you found.
 
  • Like
Likes offscene and TSny
offscene said:
Homework Statement:: Starting from the metric ##ds^2=-X^2dT^2+dX^2##, using a coordinate transformation of T and X to t and x, convert it to the form ds^2=-dt^2+dx^2 (standard flat 2D metric).
Relevant Equations:: Not actually a homework problem ("confession" from example 7.3 in Hartle's book for Gravity and GR) and no other relevant equations I can think of besides the standard chain rule but thought that this was the most fitting place to ask.

I started by expanding ##dx## and ##dt## using chain rule:

But after matching the coefficients to the original ##ds^2##, I am unable to solve the equations to come up with the right transformation and was wondering if anyone could point me in the right direction/show me.

Using the chain rule in ds^2 = -dt^2 + dx^2 = -X^2 dT^2 + dX^2 you should find <br /> \begin{split}<br /> \left(\frac{\partial x}{\partial X}\right)^2 - \left(\frac{\partial t}{\partial X}\right)^2 &amp;= 1 \\<br /> \left(\frac{\partial t}{\partial T}\right)^2 - \left(\frac{\partial x}{\partial T}\right)^2 &amp;= X^2 \\<br /> \frac{\partial t}{\partial X}\frac{\partial t}{\partial T} - \frac{\partial x}{\partial T}\frac{\partial x}{\partial X} &amp;= 0\end{split} Now the first two equations are satisfied by setting <br /> \begin{split}<br /> \frac{\partial x}{\partial X} = \cosh \zeta \quad \frac{\partial t}{\partial X} &amp;= \sinh \zeta \\<br /> \frac{\partial t}{\partial T} = X\cosh \eta \quad \frac{\partial x}{\partial T} &amp;= X\sinh \eta \end{split}<br /> due to the identify \cosh^2 u - \sinh^2 u = 1. At present \eta and \zeta are unknown functions of X and T, but the third equation gives <br /> X (\sinh \zeta \cosh \eta - \cosh \zeta \sinh \eta) = X \sinh(\zeta - \eta) = 0 so that \zeta = \eta. The problem is now reduced to finding \eta. We can either find by inspection that \eta = T will work, or we can use equality of mixed partials to find that <br /> \begin{split}<br /> \frac{\partial^2 t}{\partial X\,\partial T} - \frac{\partial^2 t}{\partial T\,\partial X} = <br /> \left(1 - \frac{\partial \eta}{\partial T}\right) \sinh \eta + X\frac{\partial \eta}{\partial X} \cosh \eta &amp;= 0 \\<br /> \frac{\partial^2 x}{\partial X\,\partial T} - \frac{\partial^2 x}{\partial T\,\partial X} = <br /> \left(1 - \frac{\partial \eta}{\partial T}\right) \cosh \eta + X\frac{\partial \eta}{\partial X} \sinh \eta &amp;= 0<br /> \end{split} and solving for the partial derivatives we find <br /> 1 - \frac{\partial \eta}{\partial T} = 0 = X\frac{\partial \eta}{\partial X}.
 
Thread 'Need help understanding this figure on energy levels'
This figure is from "Introduction to Quantum Mechanics" by Griffiths (3rd edition). It is available to download. It is from page 142. I am hoping the usual people on this site will give me a hand understanding what is going on in the figure. After the equation (4.50) it says "It is customary to introduce the principal quantum number, ##n##, which simply orders the allowed energies, starting with 1 for the ground state. (see the figure)" I still don't understand the figure :( Here is...
Thread 'Understanding how to "tack on" the time wiggle factor'
The last problem I posted on QM made it into advanced homework help, that is why I am putting it here. I am sorry for any hassle imposed on the moderators by myself. Part (a) is quite easy. We get $$\sigma_1 = 2\lambda, \mathbf{v}_1 = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} \sigma_2 = \lambda, \mathbf{v}_2 = \begin{pmatrix} 1/\sqrt{2} \\ 1/\sqrt{2} \\ 0 \end{pmatrix} \sigma_3 = -\lambda, \mathbf{v}_3 = \begin{pmatrix} 1/\sqrt{2} \\ -1/\sqrt{2} \\ 0 \end{pmatrix} $$ There are two ways...
Back
Top