What is the value of the cosmic gravitational potential?

[Vincentius]

Hi, my question regards the average (cosmic) gravitational potential and its variation over space. I have not been successful finding a good reference on this subject, other than phi=-c^2. The background gravitational potential on Earth is not necessarily equal to the average cosmic potential. After all, we live relatively close to the galactic center, so one would expect the intergalactic potential to be weaker. Can anyone help me out?
Thanks

 

[Jonathan Scott]

Hi, my question regards the average (cosmic) gravitational potential and its variation over space. I have not been successful finding a good reference on this subject, other than phi=-c^2. The background gravitational potential on Earth is not necessarily equal to the average cosmic potential. After all, we live relatively close to the galactic center, so one would expect the intergalactic potential to be weaker. Can anyone help me out?
Thanks

Newtonian gravity is a reasonable approximation at the galactic level. In dimensionless terms (potential energy per rest energy), the Newtonian potential due to a collection of masses is simply the sum of -Gm/rc² for all the relevant masses and their distances. Even for a whole galaxy, this is only something like a few parts in a million (for which you can easily calculate an approximation by looking up estimates of the mass of the Milky Way and the distance of the solar system from the center). The potential in this form is equal to the fractional change in time rate, so the ratio of time rates between two locations is like the ratio of terms of the form (1-Gm/rc²).

If you try to extend Newtonian gravity to the whole universe, it doesn't work, because the sum of the potential due to every galaxy in the universe is of order 1 and the Newtonian approximation is not valid. This is of course quite a coincidence and suggests that there may be an exact connection between G and the distribution of mass in the universe, which has led to various alternative gravity theories based on the idea that "everything is relative", usually linked to "Mach's Principle". Such theories typically imply that G must vary slightly, but such variation is incompatible with General Relativity and experimental results appear to rule out any significant variation.

 

[Naty1]

After all, we live relatively close to the galactic center,

you better double check that...I thought we were on one of the outer spirals...

found a description:

The Milky Way is a barred spiral galaxy 100,000–120,000 light-years in diameter containing 200–400 billion stars. It may contain at least as many planets.[17][18] The Solar System is located within the disk, around two thirds of the way out from the Galactic Center, on the inner edge of a spiral-shaped concentration of gas and dust called the Orion–Cygnus Arm.

http://en.wikipedia.org/wiki/Milky_Way_Galaxy

 

[Vincentius]

By "relatively" I mean as compared with an "average" position in space, i.e. likely an intergalactic position. I am looking for the effect of matter distribution on local gravitational potentials. I don't think there are simple ways to measure remote potentials other than by gravitational redshift. Stars near the galactic center are often redder than at the outskirts. This might be a clue to huge mass concentrations at the core? The reason behind my question is that I wish to know if galactic centers could be in fact much heavier than assumed on the basis of Newtonian dynamics. Rotation velocities don't match with theory. So one should put serious question marks behind Newtonian estimates of galaxy mass. My guess would be, yes, there is dark matter, but concentrated in the core (black holes), and yes, the theory needs modified to allow for flat rotation curves. This however likely requires that potential within the galaxy is dominated by the mass of the galaxy itself, which is not the general view.

 

[Chalnoth]

If you try to extend Newtonian gravity to the whole universe, it doesn't work,

Actually, as I understand it, you get the FRW equations just the same in Newtonian gravity (though the radiation contribution is wrong, of course). I am not certain whether this extends to perturbed FRW, but I strongly suspect it does to a pretty good approximation.

 

[Jonathan Scott]

By "relatively" I mean as compared with an "average" position in space, i.e. likely an intergalactic position. I am looking for the effect of matter distribution on local gravitational potentials.

Even if you include enough hidden mass or "dark matter" to explain the rotation curves, the difference in Newtonian gravitational potential somewhere in a galaxy relative to anywhere in intergalactic space is still a very tiny fraction. Again, you can easily calculate that.

Gravitational redshift only becomes significant when you're very close to a very large mass, and in such locations you are more likely to find accretion disks than objects in stable orbits.

There is another way of approximating Newtonian gravity which works in a way which is more compatible with relativistic concepts. This assumes that the relative time dilation is proportional to the exponential of the Newtonian potential in its dimensionless form. That is, rather than taking the time dilation to be (1-Sum(Gm/rc²)) compared at different points, it works better to take it to be exp(-Sum(Gm/rc²)). This gives the same result for most cases, but is unaffected by taking into account more distant objects as well.

 

[Vincentius]

There is another way of approximating Newtonian gravity which works in a way which is more compatible with relativistic concepts. This assumes that the relative time dilation is proportional to the exponential of the Newtonian potential in its dimensionless form. That is, rather than taking the time dilation to be (1-Sum(Gm/rc2)) compared at different points, it works better to take it to be exp(-Sum(Gm/rc2)). This gives the same result for most cases, but is unaffected by taking into account more distant objects as well.

Do you have a reference on this approach Jonathan?

 

[Jonathan Scott]

Do you have a reference on this approach Jonathan?

It's a common approach when discussing the Newtonian limit of General Relativity. It's probably in many textbooks - the only one in which I could immediately recall where to find it is Rindler "Essential Relativity" (revised second edition) equations 7.21 and 7.22.

 

[Vincentius]

Thanks!