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Boris
Boris
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I've read that the following hypersurface is a null surface which means that the normal four-vector has a length of 0. I would however like to confirm this myself.

$$ 2 M G \ln \left|\frac{r}{2 M G}-1\right| + t+r= B $$

where $B$ is a constant and the surface is located in Schwarzshild spacetime.

How do I compute the normal four-vector and its length?

I suspect I should use the gradient to get a normal four-vector but what does the gradient look like here? Is it made up of partial derivatives or covariant derivatives?

Update: use the metric tensor?
It seems like the covariant derivative (which is used to get the four-gradient) reduces to the normal partial derivative in this case since it is a scalar function. Wikipedia says that

In GR, one must use the more general metric tensor $g^{\alpha \beta}$, and the tensor covariant derivative (en.wikipedia.org)

I've read that the following hypersurface is a null surface which means that the normal four-vector has a length of 0. I would however like to confirm this myself.

$$ 2 M G \ln \left|\frac{r}{2 M G}-1\right| + t+r= B $$

where $B$ is a constant and the surface is located in Schwarzshild spacetime.

How do I compute the normal four-vector and its length?

I suspect I should use the gradient to get a normal four-vector but what does the gradient look like here? Is it made up of partial derivatives or covariant derivatives?

I've read that the following hypersurface is a null surface which means that the normal four-vector has a length of 0. I would however like to confirm this myself.

$$ 2 M G \ln \left|\frac{r}{2 M G}-1\right| + t+r= B $$

where $B$ is a constant and the surface is located in Schwarzshild spacetime.

How do I compute the normal four-vector and its length?

I suspect I should use the gradient to get a normal four-vector but what does the gradient look like here? Is it made up of partial derivatives or covariant derivatives?

Update: use the metric tensor?
It seems like the covariant derivative (which is used to get the four-gradient) reduces to the normal partial derivative in this case since it is a scalar function. Wikipedia says that

In GR, one must use the more general metric tensor $g^{\alpha \beta}$, and the tensor covariant derivative (en.wikipedia.org)

Source Link
Boris
Boris
  • 3
  • 2

Normal vector of hypersurface in Schwarzshild spacetime

I've read that the following hypersurface is a null surface which means that the normal four-vector has a length of 0. I would however like to confirm this myself.

$$ 2 M G \ln \left|\frac{r}{2 M G}-1\right| + t+r= B $$

where $B$ is a constant and the surface is located in Schwarzshild spacetime.

How do I compute the normal four-vector and its length?

I suspect I should use the gradient to get a normal four-vector but what does the gradient look like here? Is it made up of partial derivatives or covariant derivatives?