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Deutsch: Weltlinien von eingehenden und auslaufenden Photonen in Eddington Finkelstein Koordinaten. x=r (radiale Koordinate), y=t (Koordinatenzeit)
English: Worldlines of radially ingoing and outgoing light rays in Eddington Finkelstein coordinates. x=r (radial coordinate), y=t (coordinate time)
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Author Yukterez (Simon Tyran, Vienna)
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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])

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

Eddington-Finkelstein Space-Time-Diagram

Items portrayed in this file

depicts

23 November 2022

image/png

File history

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Date/TimeThumbnailDimensionsUserComment
current22:36, 29 November 2022Thumbnail for version as of 22:36, 29 November 20223,720 × 3,720 (205 KB)Yukterezpixel correction
17:58, 29 November 2022Thumbnail for version as of 17:58, 29 November 20223,720 × 3,720 (205 KB)Yukterezmore look better
16:26, 29 November 2022Thumbnail for version as of 16:26, 29 November 20223,720 × 3,720 (205 KB)Yukterezadding lightcones, though it is not nescessary in a photon trajectory diagram, it might help the layman
23:19, 25 November 2022Thumbnail for version as of 23:19, 25 November 20223,720 × 3,720 (207 KB)Yukterezhalf the lines are the best compromise
22:01, 25 November 2022Thumbnail for version as of 22:01, 25 November 20223,720 × 3,720 (224 KB)Yukterezmore world lines
13:30, 25 November 2022Thumbnail for version as of 13:30, 25 November 20223,720 × 3,720 (208 KB)Yukterezlarger resolution
23:51, 23 November 2022Thumbnail for version as of 23:51, 23 November 2022480 × 480 (18 KB)YukterezUploaded own work with UploadWizard

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