Does mass physically bend space or is time being bent?

[PeterDonis]

Question for me remains that since the proper mass is larger due to pressure should we not adjust the Schwarzschild radius accordingly in the formulas?

I know it seems like it ought to, but as equation (6) in the paper shows, the mass as a function of radial coordinate r, m(r), which is the parameter that appears in the metric and therefore determines the Schwarzschild radius, only includes a contribution from \rho, not p. I've seen this same derivation in MTW.

However, I agree that there is still a question of how to reconcile this result with the Komar mass integral, which does include a contribution from p. I think MTW at least touches on this issue; when I get a chance to pull my copy I'll take a look.

 

[A.T.]

Have you ever seen it done this way - i.e. spatial curvature added with the space-TIME curvature in discussing perihelion movement or light deflection?

When GR is used those two are not considered separately. You could try to compare Einsteins 1911 paper with his 1915 complete version of GR. If I remember correctly the 1911 prediction didn't consider spatial curvature:

http://mathpages.com/rr/s6-03/6-03.htm

The idea of bending light was revived in Einstein's 1911 paper "On the Influence of Gravitation on the Propagation of Light". Oddly enough, the quantitative prediction given in this paper for the amount of deflection of light passing near a large mass was identical to the old Newtonian prediction, d = 2m/r0. There were several attempts to measure the deflection of starlight passing close by the Sun during solar eclipses to test Einstein's prediction in the years between 1911 and 1915, but all these attempts were thwarted by cloudy skies, logistical problems, the First World War, etc. Einstein became very exasperated over the repeated failures of the experimentalists to gather any useful data, because he was eager to see his prediction corroborated, which he was certain it would be. Ironically, if any of those early experimental efforts had succeeded in collecting useful data, they would have proven Einstein wrong! It wasn't until late in 1915, as he completed the general theory, that Einstein realized his earlier prediction was incorrect, and the angular deflection should actually be twice the size he predicted in 1911.

 

[1977ub]

When GR is used those two are not considered separately. You could try to compare Einsteins 1911 paper with his 1915 complete version of GR. If I remember correctly the 1911 prediction didn't consider spatial curvature:

http://mathpages.com/rr/s6-03/6-03.htm

Well there you go. Ok.

“half of this deflection is produced by the Newtonian field of attraction, and the other half by the geometrical modification (‘curvature’) of space caused by the sun” - Einstein, 1916

http://mathpages.com/rr/s8-09/8-09.htm

 

[1977ub]

Just google "curvature of space" and see all the pages which seem unrelated to the curvature of space.
"Einstein's theory of general relativity describes space as curved, with the "curved space" being the four-dimensional space-time conceived of by Minowski. The curvature of space results in the effects of gravity. "
http://www.fi.edu/learn/case-files/einstein/curved.html

Ugh. Here's another one.

"Curvature of Space

A hallmark of gravity is that is causes the same acceleration no matter what the mass of the object. For example, a baseball and a cannon ball have very different masses, but if you drop them side-by-side, they accelerate downward at exactly the same rate. To explain this, Einstein envisioned gravity as being caused by a curvature of space. " - http://library.thinkquest.org/C0116043/generaltheory.htm?tql-iframe#formation

This seems to be conflating the two types of curvature of light caused by the Sun:

http://www.universetoday.com/38858/new-way-to-measure-curvature-of-space-could-unite-gravity-theory/

 

[A.T.]

Some call the 4D space-time-manifold a "space", which leads to confusion with the 3D physical space. The mathematical meaning of "space" is far more general than the physical one:
http://en.wikipedia.org/wiki/Space
http://en.wikipedia.org/wiki/Space_(mathematics)

 

[1977ub]

Generally these sources go on to describe how the curvature of "space" is responsible for gravity, so they're not discussing this topic. The only discussions of the curvature of space that I'm coming across are referring to Einstein's change in 1915.

 

[1977ub]

This is a related discussion. http://physics.stackexchange.com/qu...stortion-of-an-objects-diameter-at-a-distance

 

[pervect]

I know it seems like it ought to, but as equation (6) in the paper shows, the mass as a function of radial coordinate r, m(r), which is the parameter that appears in the metric and therefore determines the Schwarzschild radius, only includes a contribution from \rho, not p. I've seen this same derivation in MTW.

However, I agree that there is still a question of how to reconcile this result with the Komar mass integral, which does include a contribution from p. I think MTW at least touches on this issue; when I get a chance to pull my copy I'll take a look.

Both integrals give the same end result at infinity, but the distribution of "mass" is different.

I'm not aware of any papers on the topic - mostly one has to work this out for oneself. This leads to a lot of wrangling and tends not to convince people who can't follow the arguments.

An interesting case to consider is a relativistic gas or a photon gas enclosed in a rigid exotic matter shell which has rho=0. (It's exotic because it's tension is greater than it's density).

In the Komar formalism, the shell contributes a negative mass to the total. (The shell's base contribution is proportional to (rho+3P), and P is negative).

In the Schwarzschild m(r) formalism the shell doesn't contribute anything (because rho=0). However, if you compute the "mass" of the entire system "at infinity" both approaches give the same answer.

(Obviously they don't give the same answer inside the shell - the distribution of mass is different.)

There is a direct tie in between the Komarr mass enclosed by a surface and the surface integral of the force-at-infinity (see the thread https://www.physicsforums.com/showthread.php?t=679255 or the section in Wald on "Energy")

NOte defining the "force-at-infinity" requires a static metric, though.

There is no such direct tie with the m(r) and any sort of "force", though you can solve Einstein's equations to find the 4-accelearation (and hence the local force), or the 4-acceleration and the killing vectors (and hence the force-at-infinity - assuming the problem has killing vectors.)