Black Holes and Diagrams of Different Coordinate Systems
In General Relativity by Woodhouse there are the three following diagrams in Chapter 9 about Black Holes. Despite a (very brief) description of these diagrams in the book itself, I am struggling to understand what they exactly correspond to? Does anyone have good way of looking at how these diagrams help convey a) the properties of the particular coordinate system and b) a understanding of how black holes work in the particular coordinate system being looked at.
geometry differential-geometry mathematical-physics general-relativity
asked Dec 26 at 15:13
Permian
2,1801035
add a comment |
In General Relativity by Woodhouse there are the three following diagrams in Chapter 9 about Black Holes. Despite a (very brief) description of these diagrams in the book itself, I am struggling to understand what they exactly correspond to? Does anyone have good way of looking at how these diagrams help convey a) the properties of the particular coordinate system and b) a understanding of how black holes work in the particular coordinate system being looked at.
geometry differential-geometry mathematical-physics general-relativity
asked Dec 26 at 15:13
Permian
2,1801035
Crossposted from physics.stackexchange.com/q/450464/2451
– Qmechanic
Dec 26 at 16:39
add a comment |
In General Relativity by Woodhouse there are the three following diagrams in Chapter 9 about Black Holes. Despite a (very brief) description of these diagrams in the book itself, I am struggling to understand what they exactly correspond to? Does anyone have good way of looking at how these diagrams help convey a) the properties of the particular coordinate system and b) a understanding of how black holes work in the particular coordinate system being looked at.
geometry differential-geometry mathematical-physics general-relativity
asked Dec 26 at 15:13
Permian
2,1801035
In General Relativity by Woodhouse there are the three following diagrams in Chapter 9 about Black Holes. Despite a (very brief) description of these diagrams in the book itself, I am struggling to understand what they exactly correspond to? Does anyone have good way of looking at how these diagrams help convey a) the properties of the particular coordinate system and b) a understanding of how black holes work in the particular coordinate system being looked at.
geometry differential-geometry mathematical-physics general-relativity
geometry differential-geometry mathematical-physics general-relativity
asked Dec 26 at 15:13
Permian
2,1801035
asked Dec 26 at 15:13
Permian
2,1801035
asked Dec 26 at 15:13
Permian
2,1801035
asked Dec 26 at 15:13
Permian
2,1801035
asked Dec 26 at 15:13
Permian
2,1801035
2,1801035
Crossposted from physics.stackexchange.com/q/450464/2451
– Qmechanic
Dec 26 at 16:39
add a comment |
Crossposted from physics.stackexchange.com/q/450464/2451
– Qmechanic
Dec 26 at 16:39
Crossposted from physics.stackexchange.com/q/450464/2451
– Qmechanic
Dec 26 at 16:39
Crossposted from physics.stackexchange.com/q/450464/2451
– Qmechanic
Dec 26 at 16:39
add a comment |
1 Answer
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The radial null geodesics for the Schwarzschild metric have the form
$$
t pm [r + 2 m ln (r - 2m)] = k tag{1}
$$
where $k$ is a constant. This is a plot for different values of $k$
The plots you showed are just an attempt to represent these geodesics in three spatial coordinates. Unfortunately that cannot be done trivially, so instead Woodhouse decided to ignore one of the coordinates and draw Eq. (1) in the other two. You can do that by setting $x = rcosphi$ and $y = r sinphi$ for $0leq phi leq 2pi$. Or equivalently by rotating the curves in the figure above around the $t$ axis. This is an example for $k = 1$
This is just a fancy extension of the first figure above, but it is completely unnecessary, since the problem has spherical symmetry, and adding an extra coordinate just adds to the noise without any benefit.
answered Dec 26 at 19:08
caverac
13.5k21029
Hi what software did you use for plotting?
– magma
Dec 27 at 14:03
@magma Good old python
– caverac
Dec 27 at 14:04
add a comment |
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1 Answer
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1 Answer
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active
oldest
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votes
The radial null geodesics for the Schwarzschild metric have the form
$$
t pm [r + 2 m ln (r - 2m)] = k tag{1}
$$
where $k$ is a constant. This is a plot for different values of $k$
The plots you showed are just an attempt to represent these geodesics in three spatial coordinates. Unfortunately that cannot be done trivially, so instead Woodhouse decided to ignore one of the coordinates and draw Eq. (1) in the other two. You can do that by setting $x = rcosphi$ and $y = r sinphi$ for $0leq phi leq 2pi$. Or equivalently by rotating the curves in the figure above around the $t$ axis. This is an example for $k = 1$
This is just a fancy extension of the first figure above, but it is completely unnecessary, since the problem has spherical symmetry, and adding an extra coordinate just adds to the noise without any benefit.
answered Dec 26 at 19:08
caverac
13.5k21029
Hi what software did you use for plotting?
– magma
Dec 27 at 14:03
@magma Good old python
– caverac
Dec 27 at 14:04
add a comment |
The radial null geodesics for the Schwarzschild metric have the form
$$
t pm [r + 2 m ln (r - 2m)] = k tag{1}
$$
where $k$ is a constant. This is a plot for different values of $k$
The plots you showed are just an attempt to represent these geodesics in three spatial coordinates. Unfortunately that cannot be done trivially, so instead Woodhouse decided to ignore one of the coordinates and draw Eq. (1) in the other two. You can do that by setting $x = rcosphi$ and $y = r sinphi$ for $0leq phi leq 2pi$. Or equivalently by rotating the curves in the figure above around the $t$ axis. This is an example for $k = 1$
This is just a fancy extension of the first figure above, but it is completely unnecessary, since the problem has spherical symmetry, and adding an extra coordinate just adds to the noise without any benefit.
answered Dec 26 at 19:08
caverac
13.5k21029
Hi what software did you use for plotting?
– magma
Dec 27 at 14:03
@magma Good old python
– caverac
Dec 27 at 14:04
add a comment |
The radial null geodesics for the Schwarzschild metric have the form
$$
t pm [r + 2 m ln (r - 2m)] = k tag{1}
$$
where $k$ is a constant. This is a plot for different values of $k$
The plots you showed are just an attempt to represent these geodesics in three spatial coordinates. Unfortunately that cannot be done trivially, so instead Woodhouse decided to ignore one of the coordinates and draw Eq. (1) in the other two. You can do that by setting $x = rcosphi$ and $y = r sinphi$ for $0leq phi leq 2pi$. Or equivalently by rotating the curves in the figure above around the $t$ axis. This is an example for $k = 1$
This is just a fancy extension of the first figure above, but it is completely unnecessary, since the problem has spherical symmetry, and adding an extra coordinate just adds to the noise without any benefit.
answered Dec 26 at 19:08
caverac
13.5k21029
The radial null geodesics for the Schwarzschild metric have the form
$$
t pm [r + 2 m ln (r - 2m)] = k tag{1}
$$
where $k$ is a constant. This is a plot for different values of $k$
The plots you showed are just an attempt to represent these geodesics in three spatial coordinates. Unfortunately that cannot be done trivially, so instead Woodhouse decided to ignore one of the coordinates and draw Eq. (1) in the other two. You can do that by setting $x = rcosphi$ and $y = r sinphi$ for $0leq phi leq 2pi$. Or equivalently by rotating the curves in the figure above around the $t$ axis. This is an example for $k = 1$
This is just a fancy extension of the first figure above, but it is completely unnecessary, since the problem has spherical symmetry, and adding an extra coordinate just adds to the noise without any benefit.
answered Dec 26 at 19:08
caverac
13.5k21029
answered Dec 26 at 19:08
caverac
13.5k21029
answered Dec 26 at 19:08
caverac
13.5k21029
answered Dec 26 at 19:08
caverac
13.5k21029
13.5k21029
Hi what software did you use for plotting?
– magma
Dec 27 at 14:03
@magma Good old python
– caverac
Dec 27 at 14:04
add a comment |
Hi what software did you use for plotting?
– magma
Dec 27 at 14:03
@magma Good old python
– caverac
Dec 27 at 14:04
Hi what software did you use for plotting?
– magma
Dec 27 at 14:03
Hi what software did you use for plotting?
– magma
Dec 27 at 14:03
@magma Good old python
– caverac
Dec 27 at 14:04
@magma Good old python
– caverac
Dec 27 at 14:04
add a comment |
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Crossposted from physics.stackexchange.com/q/450464/2451
– Qmechanic
Dec 26 at 16:39