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Confused Physicist
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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: