Schwarzschild De Sitter Metric

English Version
Benutzeravatar
Yukterez
Administrator
Beiträge: 272
Registriert: Mi 21. Okt 2015, 02:16

Schwarzschild De Sitter Metric

Beitragvon Yukterez » Mo 17. Feb 2020, 21:32

Bild

Bild This is the english version.   Bild Deutschsprachige Version: hier entlangBild

The Schwarzschild De Sitter metric can be used to study the gravity of a dominant mass in an expanding universe:

Bild

The critical mass at which the black hole horizon coincides with the cosmic event horizon (the Nariai limit) is

Bild

where the radii of the horizons are derived by setting

Bild

to 0 and solving for the positive solutions for r.
Bild

The maximal mass of a black hole in a De Sitter universe with our cosmological constant that is Λ≈1e-52/m² would be ≈4.3e52kg:
              
Bild

x-axis: M (in kg), y-axis: r (in GLyr); green/blue: horizons (cosmic event horizon and black hole horizon), red: Schwarzschild radius rs
Bild

The attraction of a galaxy like ours is offset by the expansion of the universe at a distance of about 1 megaparsec, so that a test particle that is at that distance from the galaxy is attracted by it as much as it is accelerated away from it by the dark energy.

The physical distance of a test particle from the dominant mass remains constant at the equilibrium radius (while the distance in co-moving coordinates is shrinking). If the dominant mass rests relative to the background radiation, i.e. flows with the Hubbleflow, our stationary test particle can no longer also rest relative to the background radiation, since it is not flowing with the Hubble flow when it is stationary with respect to the central mass.

Equilibrium radius: the neutonian orbital velocity is set equal to the vacuum recession velocity and resolved to r:

Bild

     Bild

          Bild

               Bild
Bild

Derivation: the equilibrium radius at which a test particle remains stationary relative to the dominant mass is, with

Bild

and the resulting radial component of the four-acceleration

Bild

which is set to 0 and resolved to r, at

Bild

In a De Sitter universe, the relation between the Hubble constant and the cosmological constant is

Bild

plugging that into the solution for the equilibrium radius, we get the solution obtained above

Bild
Bild

For a circular orbit around the locally dominant mass we set



and solve for v:

Bild

which is an angular velocity of

Bild

where for brevity we use the dimensionless length and cosmological constant

Bild
Bild

The local escape velocity relative to a stationary buoy is with

Bild

or E=-pt=mc² and resolved to v

Bild

where the minimum is always greater than 0 and lies at the equilibrium radius mentioned above, where it is

Bild

For an observer who is traveling with the escape velocity, the gravitational and the kinematic time dilation balance out, so that

Bild

The frames of those local observers build the hypersurface of constant t in comoving coordinates.
Bild

Recessional velocity of free-falling test particles that arise near the equilibrium radius (in natural units of G=M=c=1 und Λ=0.01):

Bild

x-axis: r (in GM/c²), y-axis: v (in c); red: recessional velocity relative to stationary Fidos, green: escape velocity. Λ=c⁴/G²/M²/100=1/rs²/25
Bild

The worksheets (.nb format) can be downloaded by clicking on the respective image. Tensors, scalars and equations of motion: in Droste and Null coordinates, related topics: click here and here, next chapter: KNdS
Bild
by Simon Tyran, Vienna @ minds || gab || wikipedia || stackexchange || License: CC-BY 4. If images don't load: [ctrl]+[F5]Bild

Zurück zu „Yukterez Notepad“

Wer ist online?

Mitglieder in diesem Forum: 0 Mitglieder und 2 Gäste