File:Schwarzschild-Droste-Green-Lightcones.png
![File:Schwarzschild-Droste-Green-Lightcones.png](https://upload.wikimedia.org/wikipedia/commons/thumb/5/52/Schwarzschild-Droste-Green-Lightcones.png/600px-Schwarzschild-Droste-Green-Lightcones.png?20221127043326)
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Summary
editDescriptionSchwarzschild-Droste-Green-Lightcones.png |
Deutsch: Weltlinien von eingehenden und auslaufenden Photonen in Schwarzschild Droste Koordinaten. x=r (radiale Koordinate), y=t (Koordinatenzeit). Einige Lichtkegel sind in grün markiert.
English: Worldlines of radially ingoing and outgoing light rays in Schwarzschild Droste coordinates. x=r (radial coordinate), y=t (coordinate time). Some light cones are marked in green. |
Date | |
Source | Own work → Link |
Author | Yukterez (Simon Tyran, Vienna) |
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Photon Worldlines (v=±1, E=√[1-2/r₀])
editFree Falling Worldlines (v=±√[2/r], E=1)
editAccelerated Worldlines (v=±2/r, E=1/√[1+2/r])
editStream 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)
editLocal Observers
editIn 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 (relative to local observers stationary with respect to the black hole), while in Eddington Finkelstein coordinates they are accelerating to the squared raindrop velocity
, which they achieve by a proper acceleration of
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
, which is infinite at
.
In SD and GP coordinates, ingoing and outgoing worldlines at terminate with infinite coordinate velocity
(therefore around
they are depicted as horizontal worldlines on the spacetime diagrams), while in EF coordinates they arrive with
, which holds for timelike and lightlike geodesics (they all have
at
on the diagrams). The local velocity of photons relative to other local infalling test particles and the singularity is
though all the way, while the velocity of timelike test particles goes to
relative to the singularity.
Equations
editA1
editWith the Schwarzschild Droste line element
we get for lightlike radial paths
therefore the time by radius is
A2
editWith the Gullstrand Painlevé line element
we get for lightlike radial paths
therefore the time by radius is
for ingoing, and for outgoing rays
A3
editWith the Eddington Finkelstein line element
we get for lightlike radial paths
therefore the time by radius is
for ingoing, and for outgoing rays
B1
editFor the escape velocity we set and for photons
, then solve for
.
In Droste coordinates we get
for the free falling worldlines with the positive and negative escape velocities.
The local velocity relative to the stationary observers is
so the time by radius is
B2
editIn Raindrop coordinates we get
which gives us
B3
editIn ingoing Eddington Finkelstein coordinates we get
therefore the time by radius is
for ingoing geodesics, and for outgoing ones
C1
editWith the Schwarzschild Droste line element we get for the local velocity of :
C2
editWith the Gullstrand Painlevé line element we get
C3
editWith the Eddington Finkelstein line element
we get for the local velocity of :
D1
editThe vectors of the ingoing null conguences in Schwarzschild Droste coordinates are
D2
editThe vectors of the outgoing null conguences in Schwarzschild Droste coordinates are
D3
editThe vectors of free falling worldlines with the negative and positive escape velocity in Eddington Finkelstein coordinates are
E1
editHere we simply have .
E2
editFor the Schwarzschild Droste timelines in Raindrop coordinates we have
E3
editIn Eddington Finkelstein coordinates the Schwarzschild Droste bookkeeper timelines are given by
Units
editNatural units of are used. Code and other coordinates: Source
Common misconceptions in other diagrams
edit![This illustration got it wrong](https://upload.wikimedia.org/wikipedia/commons/thumb/5/50/SchwarzschildCoordinates1D.svg/240px-SchwarzschildCoordinates1D.svg.png)
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) was just mirrored into the
region, which looks similar to the real thing at first sight, but this is not the way to do it since the
when the radius approaches
. 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
(which goes to
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).
File history
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Date/Time | Thumbnail | Dimensions | User | Comment | |
---|---|---|---|---|---|
current | 04:33, 27 November 2022 | ![]() | 3,720 × 3,720 (364 KB) | Yukterez (talk | contribs) | making smaller, but more lightcones and adding arrows for the direction of the affine parameter |
16:51, 25 November 2022 | ![]() | 3,720 × 3,720 (315 KB) | Yukterez (talk | contribs) | Uploaded own work with UploadWizard |
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File usage on Commons
The following 15 pages use this file:
- File:Accelerated-Worldlines-Eddington-Finkelstein.png
- File:Accelerated-Worldlines-Gullstrand-Painleve.png
- File:Accelerated-Worldlines-Schwarzschild-Droste.png
- File:Curves-of-constant-Schwarzschild-time-in-Eddington-Finkelstein-coordinates.png
- File:Curves-of-constant-Schwarzschild-time-in-Gullstrand-Painleve-coordinates.png
- File:Eddington-Finkelstein-Freefall-Diagram.png
- File:Eddington-Finkelstein-Lightcone-Diagram.png
- File:Gullstrand-Painleve-Freefall-Diagram.png
- File:Gullstrand-Painleve-Lightcone-Diagram.png
- File:Ingoing-Eddington-Finkelstein-Space-Time-Vectors-Of-Free-Falling-Worldlines.png
- File:Lines-of-constant-Schwarzschild-time-in-Schwarzschild-Droste-coordinates.png
- File:Schwarzschild-Droste-Freefall-Diagram.png
- File:Schwarzschild-Droste-Lightcone-Diagram.png
- File:Schwarzschild-Droste-Space-Time-Vectors-Of-Ingoing-Null-Congruences.png
- File:Schwarzschild-Droste-Space-Time-Vectors-Of-Outgoing-Null-Congruences.png
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- Usage on en.wikipedia.org
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