- #1

- 14

- 0

Hi, I have the following problem:

Given the 5-D generalization of the Schwarszschild solution with line element:

[tex]ds^2=-\Bigg(1-\frac{r^2_+}{r^2}\Bigg)dt^2+\Bigg(1-\frac{r^2_+}{r^2}\Bigg)^{-1}dr^2+r^2[d\chi^2+\sin^2(\chi)(d\theta^2+\sin^2(\theta)d\phi^2)][/tex]

where ##r_+## is a positive constant. An observer falls radially starting from rest at ##r=10r_+##. How much time elapses on their clock before they hit the singularity at ##r=0##?

MY ATTEMPT HAS BEEN:

Using the Lemaitre coordinates ##\tau##, ##\rho## to eliminate the singularity of ##ds^2## at ##r_+##:

[tex]d\tau=dt+\frac{r_+}{r}\frac{dr}{1-\frac{r^2_+}{r^2}}[/tex]

[tex]d\rho=dt+\frac{r}{r_+}\frac{dr}{1-\frac{r_+^2}{r^2}}\quad\quad\quad (1)[/tex]

we have the following line element where the singularity at ##r_+## is removed:

[tex]ds^2=d\tau^2-\frac{r_+}{r}d\rho^2-r^2(d\theta^2+\sin^2(\theta)d\phi^2)[/tex]

where ##r=\sqrt{2(\rho-\tau)r_+}##, which is obtained by integrating [tex]d\rho-d\tau=\frac{r}{r_+}dr[/tex].

For a free falling body, ##d\rho=0##, and equation (1) gives:

[tex]dt=-\frac{r}{r_+}\frac{1}{1-\frac{r_+}{r}}dr[/tex]

Integrating this equation from ##r=10r_+## to ##r=0## should give me the time the problem asks for:

[tex]\Delta\tau=-\int_{10r_+}^0 \frac{r}{r_+}\frac{1}{1-\frac{r_+}{r}}dr[/tex]

Is this correct?

Thanks!

Given the 5-D generalization of the Schwarszschild solution with line element:

[tex]ds^2=-\Bigg(1-\frac{r^2_+}{r^2}\Bigg)dt^2+\Bigg(1-\frac{r^2_+}{r^2}\Bigg)^{-1}dr^2+r^2[d\chi^2+\sin^2(\chi)(d\theta^2+\sin^2(\theta)d\phi^2)][/tex]

where ##r_+## is a positive constant. An observer falls radially starting from rest at ##r=10r_+##. How much time elapses on their clock before they hit the singularity at ##r=0##?

MY ATTEMPT HAS BEEN:

Using the Lemaitre coordinates ##\tau##, ##\rho## to eliminate the singularity of ##ds^2## at ##r_+##:

[tex]d\tau=dt+\frac{r_+}{r}\frac{dr}{1-\frac{r^2_+}{r^2}}[/tex]

[tex]d\rho=dt+\frac{r}{r_+}\frac{dr}{1-\frac{r_+^2}{r^2}}\quad\quad\quad (1)[/tex]

we have the following line element where the singularity at ##r_+## is removed:

[tex]ds^2=d\tau^2-\frac{r_+}{r}d\rho^2-r^2(d\theta^2+\sin^2(\theta)d\phi^2)[/tex]

where ##r=\sqrt{2(\rho-\tau)r_+}##, which is obtained by integrating [tex]d\rho-d\tau=\frac{r}{r_+}dr[/tex].

For a free falling body, ##d\rho=0##, and equation (1) gives:

[tex]dt=-\frac{r}{r_+}\frac{1}{1-\frac{r_+}{r}}dr[/tex]

Integrating this equation from ##r=10r_+## to ##r=0## should give me the time the problem asks for:

[tex]\Delta\tau=-\int_{10r_+}^0 \frac{r}{r_+}\frac{1}{1-\frac{r_+}{r}}dr[/tex]

Is this correct?

Thanks!

Last edited: