Kerr_Newman_De_Sitter_(KNdS)_Horizons_&_Ergospheres.gif (620 × 464 pixels, file size: 1.79 MB, MIME type: image/gif, looped, 156 frames, 17 s)
![]() | 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. |
DescriptionKerr Newman De Sitter (KNdS) Horizons & Ergospheres.gif |
English: The horizons and ergosheres for the Kerr Newman De Sitler (KNdS) spacetime with different Λ:M ratios. The mass M, spin a and electric charge ℧ of the black hole stay constant, as does the radius of the ring singularity (r=0 → R=√[r²+a²]=a), while the cosmological constant Λ=3H² is the animation parameter. All numbers are in natural dimensionless units of G=M=c=kₑ=1. |
Date | |
Source | Own work, Code: Link |
Author | Yukterez (Simon Tyran, Vienna) |
Other versions |
![]() ![]() ![]() ![]() ![]() |
The
Kerr–Newman–de–Sitter metric (KNdS)
[1]
[2] is the one of the most general
stationary solutions of the
Einstein–Maxwell equations in
[1] that describes the spacetime geometry in the region surrounding an electrically charged, rotating mass embedded in an expanding universe. It generalizes the
Kerr–Newman metric by taking into account the
cosmological constant .
In (+, −, −, −)
signature and in
natural units of the KNdS metric is
[3]
[4]
[5]
[6]
with all the other , where
is the black hole's spin parameter,
its electric charge and
[7] the cosmological constant with
as the time-independent
Sitter universe#Mathematical expression Hubble parameter. The
electromagnetic 4-potential is
The
frame-dragging angular velocity is
and the local frame-dragging velocity relative to constant positions (the speed of light at the
ergosphere)
The escape velocity (the speed of light at the horizons) relative to the local corotating ZAMO (zero angular momentum observer) is
The conserved quantities in the equations of motion
where is the
four velocity,
is the test particle's
specific charge and
the
Maxwell–Faraday tensor
are the total energy
and the covariant axial
angular momentum
The
for differentiation overdot stands for differentiation by the testparticle's
proper time or the photon's
affine parameter, so
.
To get coordinates we apply the transformation
and get the metric coefficients
and all the other , with the electromagnetic
vector potential
Defining ingoing lightlike worldlines give a
light cone on a
spacetime diagram.
The horizons are at and the ergospheres at
.
This can be solved numerically or analytically. Like in the
Kerr and
Kerr–Newman metrics the horizons have constant Boyer-Lindquist
, while the ergospheres' radii also depend on the polar angle
.
This gives 3 positive solutions each (including the black hole's inner and outer horizons and ergospheres as well as the cosmic ones) and a negative solution for the space at in the
antiverse
[8]
[9] behind the
ring singularity, which is part of the probably unphysical extended solution of the metric.
With a negative (the
Anti–de–Sitter variant with an attractive cosmological constant) there are no cosmic horizon and ergosphere, only the black hole related ones.
In the Nariai limit
[10] the black hole's outer horizon and ergosphere coincide with the cosmic ones (in the
Schwarzschild–de–Sitter metric to which the KNdS reduces with that would be the case when
).
The
Ricci scalar for the KNdS metric is , and the
Kretschmann scalar
For the transformation see here and the links therein. More tensors and scalars for the KNdS metric: in Boyer Lindquist and Null coordinates, higher resolution: video, advised references: arxiv:1710.00997 & arxiv:2007.04354. More snapshots of this series can be found here, those are also under the creative commons license.
Click on a date/time to view the file as it appeared at that time.
Date/Time | Thumbnail | Dimensions | User | Comment | |
---|---|---|---|---|---|
current | 00:01, 11 September 2023 |
![]() | 620 × 464 (1.79 MB) | Yukterez | match the color of the cosmic ergosphere with the black hole one after they merge |
22:29, 22 August 2023 |
![]() | 620 × 464 (2.53 MB) | Yukterez | show the disintegration of the ergosphere | |
19:09, 18 August 2023 |
![]() | 640 × 464 (2.64 MB) | Yukterez | take a different start frame | |
20:05, 16 August 2023 |
![]() | 640 × 464 (2.64 MB) | 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.
Unique ID of original document | xmp.did:612f8e3c-d1b0-9b4f-b96f-31e7cc565cbb |
---|---|
Software used | Adobe Photoshop 21.0 (Windows) |
Kerr_Newman_De_Sitter_(KNdS)_Horizons_&_Ergospheres.gif (620 × 464 pixels, file size: 1.79 MB, MIME type: image/gif, looped, 156 frames, 17 s)
![]() | 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. |
DescriptionKerr Newman De Sitter (KNdS) Horizons & Ergospheres.gif |
English: The horizons and ergosheres for the Kerr Newman De Sitler (KNdS) spacetime with different Λ:M ratios. The mass M, spin a and electric charge ℧ of the black hole stay constant, as does the radius of the ring singularity (r=0 → R=√[r²+a²]=a), while the cosmological constant Λ=3H² is the animation parameter. All numbers are in natural dimensionless units of G=M=c=kₑ=1. |
Date | |
Source | Own work, Code: Link |
Author | Yukterez (Simon Tyran, Vienna) |
Other versions |
![]() ![]() ![]() ![]() ![]() |
The
Kerr–Newman–de–Sitter metric (KNdS)
[1]
[2] is the one of the most general
stationary solutions of the
Einstein–Maxwell equations in
[1] that describes the spacetime geometry in the region surrounding an electrically charged, rotating mass embedded in an expanding universe. It generalizes the
Kerr–Newman metric by taking into account the
cosmological constant .
In (+, −, −, −)
signature and in
natural units of the KNdS metric is
[3]
[4]
[5]
[6]
with all the other , where
is the black hole's spin parameter,
its electric charge and
[7] the cosmological constant with
as the time-independent
Sitter universe#Mathematical expression Hubble parameter. The
electromagnetic 4-potential is
The
frame-dragging angular velocity is
and the local frame-dragging velocity relative to constant positions (the speed of light at the
ergosphere)
The escape velocity (the speed of light at the horizons) relative to the local corotating ZAMO (zero angular momentum observer) is
The conserved quantities in the equations of motion
where is the
four velocity,
is the test particle's
specific charge and
the
Maxwell–Faraday tensor
are the total energy
and the covariant axial
angular momentum
The
for differentiation overdot stands for differentiation by the testparticle's
proper time or the photon's
affine parameter, so
.
To get coordinates we apply the transformation
and get the metric coefficients
and all the other , with the electromagnetic
vector potential
Defining ingoing lightlike worldlines give a
light cone on a
spacetime diagram.
The horizons are at and the ergospheres at
.
This can be solved numerically or analytically. Like in the
Kerr and
Kerr–Newman metrics the horizons have constant Boyer-Lindquist
, while the ergospheres' radii also depend on the polar angle
.
This gives 3 positive solutions each (including the black hole's inner and outer horizons and ergospheres as well as the cosmic ones) and a negative solution for the space at in the
antiverse
[8]
[9] behind the
ring singularity, which is part of the probably unphysical extended solution of the metric.
With a negative (the
Anti–de–Sitter variant with an attractive cosmological constant) there are no cosmic horizon and ergosphere, only the black hole related ones.
In the Nariai limit
[10] the black hole's outer horizon and ergosphere coincide with the cosmic ones (in the
Schwarzschild–de–Sitter metric to which the KNdS reduces with that would be the case when
).
The
Ricci scalar for the KNdS metric is , and the
Kretschmann scalar
For the transformation see here and the links therein. More tensors and scalars for the KNdS metric: in Boyer Lindquist and Null coordinates, higher resolution: video, advised references: arxiv:1710.00997 & arxiv:2007.04354. More snapshots of this series can be found here, those are also under the creative commons license.
Click on a date/time to view the file as it appeared at that time.
Date/Time | Thumbnail | Dimensions | User | Comment | |
---|---|---|---|---|---|
current | 00:01, 11 September 2023 |
![]() | 620 × 464 (1.79 MB) | Yukterez | match the color of the cosmic ergosphere with the black hole one after they merge |
22:29, 22 August 2023 |
![]() | 620 × 464 (2.53 MB) | Yukterez | show the disintegration of the ergosphere | |
19:09, 18 August 2023 |
![]() | 640 × 464 (2.64 MB) | Yukterez | take a different start frame | |
20:05, 16 August 2023 |
![]() | 640 × 464 (2.64 MB) | 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.
Unique ID of original document | xmp.did:612f8e3c-d1b0-9b4f-b96f-31e7cc565cbb |
---|---|
Software used | Adobe Photoshop 21.0 (Windows) |