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Finding the matricial form of Brinkmann's metric

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I have the following problem: given Brinkmann's metric expressed as
$$ds^2 = du dv - \delta_{i j} dx^i dx^j - K_{i j}(u) x^i x^j du^2$$
and $i,j=1,2, $ I have to find it's matricial form; my question is the following: how I'm supposed to do that?

I know that the general expression is $$ds^2 = g_{i j} dx^i dx^j$$
but I do not see how to write $g_{ij}$.

Subsidiary question: why is this metric useful to study gravitational waves in the void?

Thanks you!

asked Aug 13 in Physics Problems by Varazak (120 points)
edited Aug 14 by Einstein
Ups Sammy I did not know it was solved at PSE... I will leave my answer anyway OK?
@JD-PM Yes you can leave your answer. We don't have any rule against posting on other sites. I am not able to check your answer, GR is not my area of expertise.

1 Answer

1 vote

The coordinate variables are: v, x, y, u

As you stated the metric is:

$$ds^2 = g_{i j} dx^i dx^j$$

The matrix has to symmetric and the value of $\delta_{i j}$ depends on i and j as follows:

If i = j, $\delta_{i j}$ = 1

If $i \neq j$, $\delta_{i j}$ = 0

Now let's find out the value of the matrix's diagonal:

1) $g_{1 1}$

$$ds^2 = g_{1 1} dv^2$$

As this factor does not appear in the given Brinkmann's metric, $g_{1 1} = 0$. Note our first variable is v.

2) $g_{2 2}$

$$ds^2 = g_{2 2} dx^2$$

This factor does appear in the given Brinkmann's metric and $\delta_{i j}$ provides us with the answer, $g_{2 2} = -1$

3) $g_{3 3}$

For the same reason that $g_{2 2} = -1$, $g_{3 3} = -1$

4) $g_{4 4}$

$$ds^2 = g_{4 4} du^2$$

Thus:

$$g_{4 4} = -K_{i j}x^ix^j$$

You can obtain the rest of the matrix proceeding like shown. To illustrate an example different from the diagonal:

5) $g_{1 4}$

$$ds^2 = g_{1 4} dvdu$$

Because of the matrix symmetry we know:

$$g_{1 4} = g_{4 1}$$

As the coefficient of $du dv$ is 1 we know:

$$g_{1 4} + g_{4 1} = 1$$

Therefore:

$$g_{1 4} = g_{4 1} = \frac{1}{2}$$

The entire matrix:

PD: I suggest if you want to learn more about metrics, you check the lectures of Leonard Susskind on youtube, they are great.

answered Aug 19 by JD_PM (490 points)
edited Aug 20 by sammy gerbil
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