Solving Metric Tensor Problems: My Attempt at g_μν for (2)

In summary, the conversation discusses a matrix representation of the metric for two different spacetimes. The first spacetime has a non-diagonal metric and the second has a diagonal metric. The question asks if one of these spacetimes can describe Minkowski spacetime. By setting up the variables in the appropriate way and making a change of variables, it is found that the second spacetime reduces to Minkowski spacetime if ##g_{20} + g_{02} = 1##.
  • #1
WWCY
479
12
Homework Statement
Derive the metric tensors for the following spacetimes, need help with (1)
Relevant Equations
##ds^2 = g_{\mu \nu} dX^{\mu} dX^{\nu}##
Screenshot 2019-08-26 at 4.17.38 PM.png


My attempt at ##g_{\mu \nu}## for (2) was
\begin{pmatrix}
-(1-r^2) & 0 & 0 & 0 \\ 0 &\frac{1}{1-r^2} & 0 & 0 \\ 0 & 0 & r^2 & 0 \\ 0 & 0 & 0 & r^2 \sin^2(\theta)
\end{pmatrix}

and the inverse is the reciprocal of the diagonal elements.

For (1) however, I can't even think of how to write the vector ##X^{\mu}##; what exactly are ##U,V##?

Also, what does the question mean by "one of them could describe Minkowski spacetime"? At first glance, the metric tensor for (1) is non-diagonal, which I think rules it out. The metric for (2) is diagonal, and appears to approach the Minkowski metric in the small ##r## limit, which I'm guessing is the answer.

Thanks in advance!
 
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  • #3
Cross-terms in the interval means off-diagonal terms in the metric :)
 
  • #4
Thanks for the responses!

So by setting ##X^{\mu} = (U = y-t , x, V = y+t, z)## and expanding according to the line element expansion given above, I find that this form of spacetime reduces to Minkowski spacetime if ##g_{20} + g_{02} = 1##, is this right?
 
  • #5
WWCY said:
Thanks for the responses!

So by setting ##X^{\mu} = (U = y-t , x, V = y+t, z)## and expanding according to the line element expansion given above, I find that this form of spacetime reduces to Minkowski spacetime if ##g_{20} + g_{02} = 1##, is this right?
First write the 4x4 matrix in the variables x, z, U,V. (Hint: dU dV = 1/2 dU dV + 1/2 dV dU).
After that only, make the change of variables and then write the matrix in the variables x,y,z,t.
 

Related to Solving Metric Tensor Problems: My Attempt at g_μν for (2)

1. What is a metric tensor and why is it important in solving problems in physics?

A metric tensor is a mathematical object used to describe the geometry of a space. It is important in physics because it allows us to define distances and angles in curved spaces, which is necessary for understanding phenomena such as gravity.

2. How do you calculate the metric tensor for a specific space?

The metric tensor can be calculated by using the line element equation, which relates the metric tensor to the coordinates of the space. Alternatively, it can also be calculated by using the Christoffel symbols, which represent the connection between different points in a space.

3. What are the main steps involved in solving metric tensor problems?

The main steps involved in solving metric tensor problems are: understanding the physical problem and the space in which it is occurring, choosing an appropriate coordinate system, calculating the metric tensor using the line element or Christoffel symbols, and using the metric tensor to solve the problem by finding distances, angles, or other relevant quantities.

4. Can you explain the difference between a covariant and contravariant metric tensor?

A covariant metric tensor is used to measure distances and angles between vectors in the same tangent space, while a contravariant metric tensor is used to measure distances and angles between vectors in different tangent spaces. In other words, the covariant metric tensor represents the "size" of a vector in a specific direction, while the contravariant metric tensor represents the "size" of a vector as it is transformed into different coordinate systems.

5. Are there any common mistakes to avoid when solving metric tensor problems?

One common mistake is using the wrong coordinate system or metric tensor for a specific problem. It is important to carefully choose the appropriate coordinate system and metric tensor based on the geometry of the space and the physical problem being studied. Another mistake is not correctly interpreting the results obtained from solving the problem using the metric tensor, as the values may differ from what is expected in a Euclidean space.

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