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DescriptionSchwarzschild-Droste-Space-Time-Vectors-Of-Outgoing-Null-Congruences.png |
Deutsch: Vektorplot der Schwarzschild Raumzeit in Schwarzschild Droste Koordinaten. Ausgehende Photonen (v=+c). x=r, y=t |
Date | |
Source | Own work → Link |
Author | Yukterez (Simon Tyran, Vienna) |
Other versions |
![]() |
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.
With the Schwarzschild Droste line element
we get for lightlike radial paths
therefore the time by radius is
With the Gullstrand Painlevé line element
we get for lightlike radial paths
therefore the time by radius is
for ingoing, and for outgoing rays
With the Eddington Finkelstein line element
we get for lightlike radial paths
therefore the time by radius is
for ingoing, and for outgoing rays
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
In Raindrop coordinates we get
which gives us
In ingoing Eddington Finkelstein coordinates we get
therefore the time by radius is
for ingoing geodesics, and for outgoing ones
With the Schwarzschild Droste line element we get for the local velocity of :
With the Gullstrand Painlevé line element we get
With the Eddington Finkelstein line element
we get for the local velocity of :
The vectors of the ingoing null conguences in Schwarzschild Droste coordinates are
The vectors of the outgoing null conguences in Schwarzschild Droste coordinates are
The vectors of free falling worldlines with the negative and positive escape velocity in Eddington Finkelstein coordinates are
Here we simply have .
For the Schwarzschild Droste timelines in Raindrop coordinates we have
In Eddington Finkelstein coordinates the Schwarzschild Droste bookkeeper timelines are given by
Natural units of are used. Code and other coordinates:
Source
Click on a date/time to view the file as it appeared at that time.
Date/Time | Thumbnail | Dimensions | User | Comment | |
---|---|---|---|---|---|
current | 12:11, 29 November 2022 |
![]() | 3,720 × 3,720 (472 KB) | Yukterez | those were the timelike, now uploading the lightlike congruences |
12:02, 29 November 2022 |
![]() | 3,720 × 3,720 (425 KB) | Yukterez | Uploaded own work with UploadWizard |
This file contains additional information, probably added from the digital camera or scanner used to create or digitize it.
If the file has been modified from its original state, some details may not fully reflect the modified file.
Software used |
---|
Original file (3,720 × 3,720 pixels, file size: 472 KB, MIME type: image/png)
![]() | This is a file from the
Wikimedia Commons. Information from its
description page there is shown below. Commons is a freely licensed media file repository. You can help. |
DescriptionSchwarzschild-Droste-Space-Time-Vectors-Of-Outgoing-Null-Congruences.png |
Deutsch: Vektorplot der Schwarzschild Raumzeit in Schwarzschild Droste Koordinaten. Ausgehende Photonen (v=+c). x=r, y=t |
Date | |
Source | Own work → Link |
Author | Yukterez (Simon Tyran, Vienna) |
Other versions |
![]() |
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.
With the Schwarzschild Droste line element
we get for lightlike radial paths
therefore the time by radius is
With the Gullstrand Painlevé line element
we get for lightlike radial paths
therefore the time by radius is
for ingoing, and for outgoing rays
With the Eddington Finkelstein line element
we get for lightlike radial paths
therefore the time by radius is
for ingoing, and for outgoing rays
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
In Raindrop coordinates we get
which gives us
In ingoing Eddington Finkelstein coordinates we get
therefore the time by radius is
for ingoing geodesics, and for outgoing ones
With the Schwarzschild Droste line element we get for the local velocity of :
With the Gullstrand Painlevé line element we get
With the Eddington Finkelstein line element
we get for the local velocity of :
The vectors of the ingoing null conguences in Schwarzschild Droste coordinates are
The vectors of the outgoing null conguences in Schwarzschild Droste coordinates are
The vectors of free falling worldlines with the negative and positive escape velocity in Eddington Finkelstein coordinates are
Here we simply have .
For the Schwarzschild Droste timelines in Raindrop coordinates we have
In Eddington Finkelstein coordinates the Schwarzschild Droste bookkeeper timelines are given by
Natural units of are used. Code and other coordinates:
Source
Click on a date/time to view the file as it appeared at that time.
Date/Time | Thumbnail | Dimensions | User | Comment | |
---|---|---|---|---|---|
current | 12:11, 29 November 2022 |
![]() | 3,720 × 3,720 (472 KB) | Yukterez | those were the timelike, now uploading the lightlike congruences |
12:02, 29 November 2022 |
![]() | 3,720 × 3,720 (425 KB) | Yukterez | Uploaded own work with UploadWizard |
This file contains additional information, probably added from the digital camera or scanner used to create or digitize it.
If the file has been modified from its original state, some details may not fully reflect the modified file.
Software used |
---|