File:Schwarzschild-Droste-Green-Lightcones.png

Summary[edit]

Licensing[edit]

I, the copyright holder of this work, hereby publish it under the following license:

This file is licensed under the Creative Commons Attribution-Share Alike 4.0 International license.

You are free:

• to share – to copy, distribute and transmit the work
• to remix – to adapt the work

Under the following conditions:

• attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
• share alike – If you remix, transform, or build upon the material, you must distribute your contributions under the same or compatible license as the original.

https://creativecommons.org/licenses/by-sa/4.0CC BY-SA 4.0 Creative Commons Attribution-Share Alike 4.0 truetrue

Photon Worldlines (v=±1, E=√[1-2/r₀])[edit]

• 
  A1) Schwarzschild Droste
• 
  A2) Gullstrand Painlevé
• 
  A3) Eddington Finkelstein

Free Falling Worldlines (v=±√[2/r], E=1)[edit]

• 
  B1) Schwarzschild Droste
• 
  B2) Gullstrand Painlevé
• 
  B3) Eddington Finkelstein

Accelerated Worldlines (v=±2/r, E=1/√[1+2/r])[edit]

• 
  C1) Schwarzschild Droste
• 
  C2) Gullstrand Painlevé
• 
  C3) Eddington Finkelstein

Stream Plots (v=±1 & v=-√[2/r])[edit]

• 
  D1) SD, ingoing lightlike vectors
• 
  D2) SD, outgoing lightlike vectors
• 
  D3) EF, ingoing timelike vectors

Curves of constant bookkeeper time (t=constant)[edit]

• 
  E1) Schwarzschild Droste
• 
  E2) Gullstrand Painlevé
• 
  E3) Eddington Finkelstein

Local Observers[edit]

In Gullstrand Painlevé coordinates the local observers (or clocks and rulers) who define the direction of the space and time axes are free falling raindrops with the negative escape velocity ${\displaystyle {\rm {v=-\surd [2/r]}}}$ (relative to local observers stationary with respect to the black hole), while in Eddington Finkelstein coordinates they are accelerating to the squared raindrop velocity ${\displaystyle {\rm {v=-2/r}}}$, which they achieve by a proper acceleration of ${\displaystyle {\rm {a=F/m=1/\surd [(r+2)^{3}r]}}}$ radially outwards, so de facto a deceleration. In the classic Schwarzschild Droste coordinates the local clocks and rulers are stationary with respect to the black hole, so they also experience a proper outward acceleration of ${\displaystyle {\rm {a=1/r^{2}/\surd [1-2/r]}}}$, which is infinite at ${\displaystyle {\rm {r=2}}}$.

In SD and GP coordinates, ingoing and outgoing worldlines at ${\displaystyle {\rm {r\to 0}}}$ terminate with infinite coordinate velocity ${\displaystyle {\rm {dr/dt\to -1/0}}}$ (therefore around ${\displaystyle {\rm {r=0}}}$ they are depicted as horizontal worldlines on the spacetime diagrams), while in EF coordinates they arrive with ${\displaystyle {\rm {dr/dt=-1}}}$, which holds for timelike and lightlike geodesics (they all have ${\displaystyle 45\mathrm {{}^{\circ }} }$ at ${\displaystyle {\rm {r\to 0}}}$ on the diagrams). The local velocity of photons relative to other local infalling test particles and the singularity is ${\displaystyle {\rm {v=\pm 1}}}$ though all the way, while the velocity of timelike test particles goes to ${\displaystyle {\rm {v\to -\infty }}}$ relative to the singularity.

Equations[edit]

A1[edit]

With the Schwarzschild Droste line element

${\displaystyle {\rm {ds^{2}=g_{tt}\ dt^{2}+g_{rr}\ dr^{2}=(1-2/r)\ dt^{2}-dr/(1-2/r)=0}}}$

we get for lightlike radial paths

${\displaystyle {\rm {{\bar {r}}=dr/dt=\pm (r-2)/r}}}$

therefore the time by radius is

${\displaystyle {\rm {t=\int _{r_{1}}^{r_{2}}dr/{\bar {r}}=\pm (r_{1}+r_{2}-2\ln(r_{1}-2)+2\ln(r_{2}-2))}}}$

A2[edit]

With the Gullstrand Painlevé line element

${\displaystyle {\rm {ds^{2}=g_{tt}\ dt^{2}+g_{rr}\ dr^{2}+2g_{tr}\ dt\ dr=(1-2/r)\ dt^{2}-dr-{\sqrt {8/r}}\ dr\ dt=0}}}$

we get for lightlike radial paths

${\displaystyle {\rm {{\bar {r}}=dr/dt=-1-{\sqrt {2/r}}\ ||\ 1-{\sqrt {2/r}}}}}$

therefore the time by radius is

${\displaystyle {\rm {t=\int _{r_{1}}^{r_{2}}dr/{\bar {r}}=r_{1}-2{\sqrt {2}}{\sqrt {r_{1}}}+4\ln \left({\sqrt {r_{1}}}+{\sqrt {2}}\right)-r_{2}+2{\sqrt {2}}{\sqrt {r_{2}}}-4\ln \left({\sqrt {r_{2}}}+{\sqrt {2}}\right)}}}$

for ingoing, and for outgoing rays

${\displaystyle {\rm {t=-r_{1}-2{\sqrt {2}}{\sqrt {r_{1}}}-2\ln(r_{1})-4\ln \left(1-{\sqrt {2/r_{1}}}\right)+r_{2}+2{\sqrt {2}}{\sqrt {r_{2}}}+2\ln(r_{2})+4\ln \left(1-{\sqrt {2/r_{2}}}\right)}}}$

A3[edit]

With the Eddington Finkelstein line element

${\displaystyle {\rm {ds^{2}=g_{tt}\ dt^{2}+g_{rr}\ dr^{2}+2g_{tr}\ dt\ dr=(1-2/r)\ dt^{2}-(1+2/r)\ dr-4/r\ dr\ dt=0}}}$

we get for lightlike radial paths

${\displaystyle {\rm {{\bar {r}}=dr/dt=-1\ ||\ (r-2)/(r+2)}}}$

therefore the time by radius is

${\displaystyle {\rm {t=\int _{r_{1}}^{r_{2}}dr/{\bar {r}}=r_{1}-r_{2}}}}$

for ingoing, and for outgoing rays

${\displaystyle {\rm {t=-r_{1}-4\ln(r_{1}-2)+r_{2}+4\ln(r_{2}-2)}}}$

B1[edit]

For the escape velocity we set ${\displaystyle {\rm {E=-p_{r}=g_{tt}\ dt/ds=1}}}$ and for photons ${\displaystyle {\rm {ds=0}}}$, then solve for ${\displaystyle {\rm {dr/dt}}}$.

In Droste coordinates we get

${\displaystyle {\rm {dr/dt=\pm (1-2/r){\sqrt {2/r}}}}}$

for the free falling worldlines with the positive and negative escape velocities.

The local velocity relative to the stationary observers is

${\displaystyle {\rm {v={\frac {dr}{dt}}\ {\sqrt {-{\frac {g_{rr}}{g_{tt}}}}}}}}$

so the time by radius is

${\displaystyle {\rm {\pm t=-{\frac {1}{3}}{\sqrt {2}}\left(r_{1}^{3/2}+6{\sqrt {r_{1}}}-{\sqrt {r_{2}}}(r_{2}+6)\right)+4\ arctanh{\sqrt {2/r_{1}}}-4\ arctanh{\sqrt {2/r_{2}}}}}}$

B2[edit]

In Raindrop coordinates we get

${\displaystyle {\rm {{\frac {dr}{dt}}={\frac {r-2}{r+2}}\ {\sqrt {\frac {2}{r}}}\ ||-{\sqrt {\frac {2}{r}}}}}}$

which gives us

${\displaystyle {\rm {t={\frac {1}{3}}{\sqrt {2}}\left(r_{1}^{3/2}-r_{2}^{3/2}\right)\ ||\ -{\frac {1}{3}}{\sqrt {2}}\left(r_{1}^{3/2}+12{\sqrt {r_{1}}}-{\sqrt {r_{2}}}(r_{2}+12)\right)+8\ arctanh{\sqrt {2/r_{1}}}-8\ arctanh{\sqrt {2/r_{2}}}}}}$

B3[edit]

In ingoing Eddington Finkelstein coordinates we get

${\displaystyle {\rm {{\frac {dr}{dt}}={\frac {{\sqrt {2}}\ (r-2)}{{\sqrt {8}}\pm r^{3/2}}}}}}$

therefore the time by radius is

${\displaystyle {\rm {t={\frac {1}{3}}{\sqrt {2}}\left(r_{1}^{3/2}+6{\sqrt {r_{1}}}-{\sqrt {r_{2}}}(r_{2}+6)\right)+2\ln \left({\frac {r_{2}-2}{r_{1}-2}}\right)-4\ arctanh{\sqrt {2/r_{1}}}+4\ arctanh{\sqrt {2/r_{2}}}}}}$

for ingoing geodesics, and for outgoing ones

${\displaystyle {\rm {t=-{\frac {1}{3}}{\sqrt {2}}\left(r_{1}^{3/2}+6{\sqrt {r_{1}}}-{\sqrt {r_{2}}}(r_{2}+6)\right)+2\ln \left({\frac {r_{2}-2}{r_{1}-2}}\right)+4\ arctanh{\sqrt {2/r_{1}}}-4\ arctanh{\sqrt {2/r_{2}}}}}}$

C1[edit]

With the Schwarzschild Droste line element we get for the local velocity of ${\displaystyle {\rm {v=\pm v_{e}^{2}}}}$:

${\displaystyle {\rm {dr/dt=\pm (2r-4)/r}}}$

C2[edit]

With the Gullstrand Painlevé line element we get

${\displaystyle {\rm {dr/dt={\frac {4-2r}{r^{2}-{\sqrt {8r}}}}\ ||\ {\frac {2r-4}{r^{2}+{\sqrt {8r}}}}}}}$

C3[edit]

With the Eddington Finkelstein line element

${\displaystyle {\rm {ds^{2}=g_{tt}\ dt^{2}+g_{rr}\ dr^{2}+2g_{tr}\ dt\ dr=(1-2/r)\ dt^{2}-(1+2/r)\ dr-4/r\ dr\ dt}}}$

we get for the local velocity of ${\displaystyle {\rm {v=\pm v_{e}^{2}}}}$:

${\displaystyle {\rm {dr/dt=-{\frac {2}{r+2}}\ ||\ {\frac {2(r-2)}{r^{2}+4}}}}}$

D1[edit]

The vectors of the ingoing null conguences in Schwarzschild Droste coordinates are

${\displaystyle {\rm {\{dt,\ dr\}=sign(r-2)\ \{1,\ 2/r-1\}}}}$

D2[edit]

The vectors of the outgoing null conguences in Schwarzschild Droste coordinates are

${\displaystyle {\rm {\{dt,\ dr\}=\{1,\ 1-2/r\}}}}$

D3[edit]

The vectors of free falling worldlines with the negative and positive escape velocity in Eddington Finkelstein coordinates are

${\displaystyle {\rm {\{dt,\ dr\}=\{1,\ {\sqrt {2}}(r-2)/({\sqrt {8}}\pm r^{3/2})\}}}}$

E1[edit]

Here we simply have ${\displaystyle {\rm {t\to t}}}$.

E2[edit]

For the Schwarzschild Droste timelines in Raindrop coordinates we have

${\displaystyle {\rm {t\to t-\left({\sqrt {8}}\left({\sqrt {r_{2}}}-{\sqrt {r_{1}}}\right)+4\ arctanh{\sqrt {2/r_{1}}}-4\ arctanh{\sqrt {2/r_{2}}}\right)}}}$

E3[edit]

In Eddington Finkelstein coordinates the Schwarzschild Droste bookkeeper timelines are given by

${\displaystyle {\rm {t\to t-\ln \left({\frac {r_{2}^{2}-4}{r_{1}^{2}-4}}\right)}}}$

Units[edit]

Natural units of ${\displaystyle {\rm {G=M=c=1}}}$ are used. Code and other coordinates: Source

Common misconceptions in other diagrams[edit]

With Google search and also on some Wikipedia articles one finds images like the one shown on the left, where the part to the right (i.e. above) ${\displaystyle {\rm {r>2}}}$ was just mirrored into the ${\displaystyle {\rm {r<2}}}$ region, which looks similar to the real thing at first sight, but this is not the way to do it since the ${\displaystyle {\rm {dt/dr\to -0}}}$ when the radius approaches ${\displaystyle {\rm {r\to 0}}}$. The world lines should be horizontal around the singularity in this coordinates, which you don't get when the outside region is just mirrored into the region behind the horizon. One often finds images like this, where the author either simply copied such an artist's impression, or found some other diagram for different metrics like the "Chern Simons metric in Schwarzschild-like coordinates" or whatever Google spits out when searching for a diagram of the original Schwarzschild metric without looking at the fine print. This even happens at otherwise serious references, like the one given in the link of the illustration in question on the left, which shows the correct equation for ${\displaystyle {\rm {dt/dr}}}$ (which goes to ${\displaystyle 0}$ at the center) though, but above an incorrect illustration. With all the correct diagrams for all the different world lines in the different coordinates and the comprehensible equations on one page there should be no confusions any more.

Some references that got it right are R. L. Herman, A. Hamilton, Eigenchris, Trin Tragula (If I stumble upon more I will add them to the list).