- #1

George Keeling

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- TL;DR Summary
- On the way to Kruskal I meet null coordinates and want to know a bit more. Does the metric signature change in Schwarzschild?

On the way to Kruskal coordinates, Carroll introduces coordinates ##\left(v^\prime,u^\prime,\theta,\phi\right)## with metric equation$$

{ds}^2=-\frac{2{R_s}^3}{r}e^{-r / R_s}\left(dv^\prime du^\prime+du^\prime dv^\prime\right)+r^2{d\Omega}^2

$$

##R_s=2GM## and we're using a ##-+++## signature and the speed of light ##c=1##. The familiar Schwarzschild coordinate ##r## is used in the equation and implicitly defined by $$

u^\prime v^\prime=-\left(\frac{r}{R_s}-1\right)e^{r/R_s}

$$Carroll then writes "Both ##v^\prime## and ##u^\prime## are null coordinates, in the sense that their partial derivatives ##\partial/\partial v^\prime## and ##\partial/\partial v^\prime## are null vectors. There is nothing wrong with this ...".

I suppose that null vectors are ones that lie on the light cone in contrast to timelike and spacelike vectors. Then I have some questions:

1) I can see that in Minkowski space ##t## is timelike because it's axis goes 'up' and ##x,y,z## are spacelike because their axes go 'across'. This sounds like a tautology. Is there a more rigorous way of saying that?

2) How would I calculate ##\partial/\partial v^\prime## and ##\partial/\partial v^\prime## and show they are on a light cone?

3) The diagonal metric components in Minkowski are negative, positive, positive, positive. Does the sign tell you whether the corresponding coordinate is timelike or spacelike? And if the component vanishes, as in the metric above, does that mean the coordinate is null?

4) With the Schwarzschild metric the signs of the diagonal metric ##t## and ##r## components reverse inside the event horizon. I have read that avoiding the singularity is as impossible as avoiding old age. Is it correct to say that inside the event horizon ##t## is spacelike and ##r## timelike? How do you know that decreasing ##r## on the inside is like increasing ##t## on the outside?

5) Does the Schwarzschild metric signature become ##-+--## inside the event horizon? Is the metric signature of the ##\left(v^\prime,u^\prime,\theta,\phi\right)## system ##0\ 0++##?

Thanks!

{ds}^2=-\frac{2{R_s}^3}{r}e^{-r / R_s}\left(dv^\prime du^\prime+du^\prime dv^\prime\right)+r^2{d\Omega}^2

$$

##R_s=2GM## and we're using a ##-+++## signature and the speed of light ##c=1##. The familiar Schwarzschild coordinate ##r## is used in the equation and implicitly defined by $$

u^\prime v^\prime=-\left(\frac{r}{R_s}-1\right)e^{r/R_s}

$$Carroll then writes "Both ##v^\prime## and ##u^\prime## are null coordinates, in the sense that their partial derivatives ##\partial/\partial v^\prime## and ##\partial/\partial v^\prime## are null vectors. There is nothing wrong with this ...".

I suppose that null vectors are ones that lie on the light cone in contrast to timelike and spacelike vectors. Then I have some questions:

1) I can see that in Minkowski space ##t## is timelike because it's axis goes 'up' and ##x,y,z## are spacelike because their axes go 'across'. This sounds like a tautology. Is there a more rigorous way of saying that?

2) How would I calculate ##\partial/\partial v^\prime## and ##\partial/\partial v^\prime## and show they are on a light cone?

3) The diagonal metric components in Minkowski are negative, positive, positive, positive. Does the sign tell you whether the corresponding coordinate is timelike or spacelike? And if the component vanishes, as in the metric above, does that mean the coordinate is null?

4) With the Schwarzschild metric the signs of the diagonal metric ##t## and ##r## components reverse inside the event horizon. I have read that avoiding the singularity is as impossible as avoiding old age. Is it correct to say that inside the event horizon ##t## is spacelike and ##r## timelike? How do you know that decreasing ##r## on the inside is like increasing ##t## on the outside?

5) Does the Schwarzschild metric signature become ##-+--## inside the event horizon? Is the metric signature of the ##\left(v^\prime,u^\prime,\theta,\phi\right)## system ##0\ 0++##?

Thanks!