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Buzz Bloom
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After learning a bit from the thread "The Dark Sky Ahead", I have tried to think through the dynamics of gravitational interactions in a simplified scenario. Unfortunately the GR math of the problem is over my head. My attempt below, using a simpler approach, leaves me quite confused.
If my analysis below is correct, this would seem to suggest that the approach that is commonly used, comparing escape velocity with Hubble velocity to determine if a system is bound with respect to the universe expansion, is wrong.
Consider an (almost) empty universe expanding exponentially with a single point mass and a test particle.
Let p be the test particle.
Let P be the point M.
Let M be the mass of P.
Let R be the distance between P and p.
Let h be the fixed constant value of the Hubble constant.
Let G be the Gravitational constant.
The questions I want to investigate are about stable configurations.
Scenario 1. p is stationary, that is, not moving with respect to P
Case (a): escape velocity = Hubble velocity
The square of the escape velocity
Question 1a:
Case (b): gravitational acceleration = Hubble acceleration
The gravitational acceleration is
The value of R for which these two accelerations are equal is
Note: Clearly Q1a and Q1b cannot both be answered "yes".Scenario 2. p is in an orbit about P.
Question 2a: Is a circular orbit possible?
Question 2b: If so, is the orbital velocity affected by the universe expansion?
Question 2c: If so, how?
The following are my guesses at answers.
Guess (2a): yes.
Guess (2b): yes.
Guess (2c):
The square of the orbital velocity with h = 0 is
Note: R2c = R1b
My guess is that the inward gravitational acceleration must equal the sum of the outward accelerations: Hubble plus centrifugal.
The analysis in Scenario 2 seem to suggest the conclusion I included at the beginning of this post.
If my analysis below is correct, this would seem to suggest that the approach that is commonly used, comparing escape velocity with Hubble velocity to determine if a system is bound with respect to the universe expansion, is wrong.
Consider an (almost) empty universe expanding exponentially with a single point mass and a test particle.
Let p be the test particle.
Let P be the point M.
Let M be the mass of P.
Let R be the distance between P and p.
Let h be the fixed constant value of the Hubble constant.
Let G be the Gravitational constant.
The questions I want to investigate are about stable configurations.
Scenario 1. p is stationary, that is, not moving with respect to P
Case (a): escape velocity = Hubble velocity
The square of the escape velocity
Vesc2 = 2 G M / R
The square of the radial velocity of p relative to P due to the expanding universeVexp2 = h2 R2
The value of R for which these two velocities squared values are equal isR1a = (2 G M / h2)1/3
Question 1a:
If p is undisturbed by any outside force, is this static configuration with R = R1a possible. (I understand that if it is, any disturbance would cause p to eventually move to infinity.)
Case (b): gravitational acceleration = Hubble acceleration
The gravitational acceleration is
Ag = G M / R2
The Hubble acceleration is what p would experience relative to P if M = 0. It isAh = h2 R
The value of R for which these two accelerations are equal is
R1b = (G M / h2)1/3
Question 1b:
Same as question (1a) except that R = R1b.
Note: Clearly Q1a and Q1b cannot both be answered "yes".Scenario 2. p is in an orbit about P.
Question 2a: Is a circular orbit possible?
Question 2b: If so, is the orbital velocity affected by the universe expansion?
Question 2c: If so, how?
The following are my guesses at answers.
Guess (2a): yes.
Guess (2b): yes.
Guess (2c):
The square of the orbital velocity with h = 0 is
Vorb2 = G M / R
The square of the radial velocity of p relative to P due to the expanding universe isVexp2 = h2 R2
The total velocity squared is the sumVtot2 = G M / R + h2 R2
The square of the escape velocity isVesc2 = 2 G M / R
Vtot = Vesc impliesG M / R = h2 R2,
and the value of R isR2c = (G M / h2)1/3
Note: R2c = R1b
My guess is that the inward gravitational acceleration must equal the sum of the outward accelerations: Hubble plus centrifugal.
G M / R2 = h2 R + Vorb2 / R
Vorb2 = G M / R - h2 R2
When Vorb = 0,Vorb2 = G M / R - h2 R2
G M / R = h2 R2
This is the same result as in Scenario 1 Case (b).The analysis in Scenario 2 seem to suggest the conclusion I included at the beginning of this post.
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