File:Curves-of-constant-Schwarzschild-time-in-Gullstrand-Painleve-coordinates.png
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Summary
DescriptionCurves-of-constant-Schwarzschild-time-in-Gullstrand-Painleve-coordinates.png |
English: x: r (reduced circumference r=U/2/π), y: t (raindrop time) |
Date | |
Source | Own work → Link |
Author | Yukterez (Simon Tyran, Vienna) |
Other versions |
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Photon Worldlines (v=±1, E=√[1-2/r₀])
Free Falling Worldlines (v=±√[2/r], E=1)
Accelerated Worldlines (v=±2/r, E=1/√[1+2/r])
Stream Plots (v=±1 & v=-√[2/r])
-
D1) SD, ingoing lightlike vectors
-
D2) SD, outgoing lightlike vectors
-
D3) EF, ingoing timelike vectors
Curves of constant bookkeeper time (t=constant)
Local Observers
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 (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
A1
With the Schwarzschild Droste line element
we get for lightlike radial paths
therefore the time by radius is
A2
With the Gullstrand Painlevé line element
we get for lightlike radial paths
therefore the time by radius is
for ingoing, and for outgoing rays
A3
With the Eddington Finkelstein line element
we get for lightlike radial paths
therefore the time by radius is
for ingoing, and for outgoing rays
B1
For 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
In Raindrop coordinates we get
which gives us
B3
In ingoing Eddington Finkelstein coordinates we get
therefore the time by radius is
for ingoing geodesics, and for outgoing ones
C1
With the Schwarzschild Droste line element we get for the local velocity of :
C2
With the Gullstrand Painlevé line element we get
C3
With the Eddington Finkelstein line element
we get for the local velocity of :
D1
The vectors of the ingoing null conguences in Schwarzschild Droste coordinates are
D2
The vectors of the outgoing null conguences in Schwarzschild Droste coordinates are
D3
The vectors of free falling worldlines with the negative and positive escape velocity in Eddington Finkelstein coordinates are
E1
Here we simply have .
E2
For the Schwarzschild Droste timelines in Raindrop coordinates we have
E3
In Eddington Finkelstein coordinates the Schwarzschild Droste bookkeeper timelines are given by
Units
Natural units of are used. Code and other coordinates: Source
Captions
Items portrayed in this file
depicts
some value
5 December 2022
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3,720 pixel
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File history
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Date/Time | Thumbnail | Dimensions | User | Comment | |
---|---|---|---|---|---|
current | 23:36, 5 December 2022 | ![]() | 3,720 × 3,720 (206 KB) | Yukterez | accidentally uploaded a version without pixel correction at r=2, now uploading the smooth one |
23:31, 5 December 2022 | ![]() | 3,720 × 3,720 (207 KB) | Yukterez | centering in with steps of 2 on the right | |
20:45, 5 December 2022 | ![]() | 3,720 × 3,720 (202 KB) | Yukterez | Uploaded own work with UploadWizard |
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