I have always thought that GR was identical to classical physics in the Newtonian limit. In the Newtonian limit one can define an energy density for a gravitational field parallel to the methods used to derive the energy density of a static electric field, resulting in a formula energy density = |g|^2/8*pi*G where g is Gm/r^2 outside a spherical mass distribution. Shouldn't this energy contribute to the space-time curvature, or is this another way to interpret the space time curvature? A classical integration of the mass, including the mass of the gravitational field above, blows up at a radius of rs/4, where rs is the Swartzchild radius. Is it a coincidence that the isotropic coordinates often used also have a singularity at r=rs/4? I am currently looking for a consistent way to resolve the fact that my classical derivation does not account for time dilation as the field strength increases. My intuition tells me that the length the object travels is contracted by a net 3/4*rs by the gravitational field, but can't show this yet. Since there is no preferred reference frame, is it valid to simply pick a convenient one? When I pick one at infinity co-moving with the central mass it appears to me that space is flowing towards the central mass with time. Is my intuition correct that the proper time is covariant, but once I pick a frame my time is contravariant? Is space time-curvature in a covariant description the same as space flowing with time in a contravariant description? I have a very limited knowledge of tensors (which might make these questions trivial) so a solid resource on this would be greatly appreciated.
If you don't take time dilation into account at all, then the energy density of a classical Newtonian gravitational field is negative (unlike in electrostatics, where like charges repel). There are big complications comparing energy distribution in GR with Newtonian gravity because of the totally different viewpoints, but we can look at the weak field approximation and try to model it in Newtonian terms. When one object is lowered (extracting the energy) to a point closer to another in GR, the resulting time dilation due to the potential caused by the other object effectively causes the same change in energy as the decrease of potential energy in Newtonian theory. However, exactly the same potential energy change also applies from the point of view of the other object. As both objects are affected by the other one, the apparent total loss of energy is TWICE the change in the potential energy of the system as a whole. One known mathematically consistent answer to this (which is apparently used in some forms of Quantum Field Theory) is to assume that the field has a positive energy density given by the usual expression, at least as seen against a Newtonian background space. The integral of this energy density over all space then exactly cancels out one copy of the potential energy, so the overall energy of the system has only changed by the usual potential energy. Note that this energy density in empty space does not show up in the usual way of looking at GR, because the field can always be transformed away by choosing an accelerated frame of reference where no work is being done on a free-falling object.
Johnathan, Thank you for your comments. I neglected the - above. The negative energy of the gravitational field can also be expected from the fact that the momentum of the field is opposed to the motion of the field, if one considers a graviton. In summary, are you saying that in the Newtonian limit of GR, the resulting time dilation due to the potential caused by the other object is effectively that of a change in velocity of the underlying space-time as predicted by the acceleration Newton's theory but within the context of SR? I believe the same factor of 2 appears if one assumes that the point on an infintesimally thin hollow sphere feels the mass of the sphere at the center of mass. Essentially that factor of two arises in the mathematics because the total energy in the field needs to be built up from zero energy and is proportional to the field itself. Thus we integrate the field with respect to itself and get a result field squared/2. I can see that this would get rid of the field locally. Does this get rid of the field globally? Isn't the negative curvature of space around an object (needed to bring the positive curvature of the mass back to flat in the Newtonian limit) the same as a negative energy distribution in that space-time?
No, I don't think I was actually saying that, but something like that is true anyway and I guess it's closely related to what I was saying (in that it explains how the kinetic energy change from SR matches the potential energy change). Yes, something like that is correct. When you build up a single object or a system of objects by bringing in mass from infinity, the potential energy of each new bit is affected by the stuff that is already there, and this results in an integral like that. In the Newtonian approximation, the scheme of assuming the time dilation based on the potential energy and the positive field energy as in my previous post works mathematically not just for two bodies but for any collection of mass. No, of course not, but it means that we cannot describe energy transfer locally. Nice image, but it doesn't work like that. I guess you're thinking of a dimple in a surface, curved upwards, being surrounded by an area which is curved downwards. Firstly, the vertical in that picture is not considered in GR; we are only concerned with the projection of that picture into the plane. Secondly, any curvature in that picture which could be locally represented by bending flat paper (like a cone or part of a cylinder) doesn't count as curvature in the GR sense caused locally by mass and energy. And thirdly, my personal, totally unofficial, opinion is that it makes more sense if space ISN'T flat at distance from a central object but is instead effectively a cone whose angle from flatness depends on the proportion of the mass of the universe enclosed within the boundaries (as a higher dimensional equivalent of Descarte's angular deficit). This has the effect that around very large collection of masses (such as galaxies) the conical effect causes an apparent additional force which appears to be similar to that in MOND.
The fundamental issue here is that the Newtonian gravitational field g is not a well-defined thing in GR. By the equivalence principle, a free-falling observer sees g=0 at any given point in spacetime. Because of this, it doesn't make sense to define an energy density using |g|^2. You can do so only in an appropriate Newtonian limit. The textbook by Carroll has a good discussion of this.
In GR it appears to me that the curvature of space-time imparts energy and momentum as seen by a distant observer, acting as g. In SR it is possible to define an inverse square distance field that is proportional to the energy density of a differential volume element. Examining a spherical mass one can calculate an energy density of this field. By shrinking this sphere from infinity and asserting that the negative energy of the field is balanced by an increase in energy of the sphere one can find where the energy of the sphere goes to infinity. This is rs/4. Now considering a much smaller bit of energy moving in such a field it is possible to determine its energy increase as it decends into the field. This increase is proportional to the mass and can be treated as a gamma for an apparent flow into the mass caused by using clocks set to a clock at infinity instead of a local clock. When I include the modifications to the gravitational force caused by the mass of the field, the gamma caused by the apparent flow into the source and the velocity of a circular orbit, then modify by dividing by (1-e^2) to account for ecentricity, I get a value for the precession of Mercury almost identical to that of GR. In fact, the first order differences in the theory are identical to GR, but higher orders vary. I am not savvy enough to determine if this is due do real differences in the theory or due to differences in measuring time. In the isotropic chart of the Swartzchild metric the modified radius also transforms rs to rs/4. Is using SR and modeling the space-time curvature of GR as a negative energy in a g field isomorphic to the isotropic coordinates? If so, this would suggest that the rs/4 value (and the isotropic radius in general) is the radius of a black hole observed if everyone set their clocks to match a theoretical clock comoving with the system barycenter but outside the gravitational fields rather than using a local proper time.
Utesfan100, you really should read a standard textbook treatment of this topic before you try to reinvent the wheel.
Can you recommend one that I could find in the library at the community college where I work? I have not found a good junior-senior level introductory treatment on tensors here. All I have is my engineering intution. Thankfully reinventing wheels is done all the time. I would hate to ride in my car with the original design.
I feel your pain -- I teach at a community college myself. But don't you have access to inter-library loan?
I can just about agree with the "momentum" part of that first sentence, but the rest of your post doesn't make much sense. (Note that a static gravitational field does not impart energy - it merely changes it between potential and kinetic). Many situations such as simple gravitational fields and centrifugal force can be handled either using the GR approach or an SR approach. Both give the same result. The relationship that potential energy is twice kinetic energy for circular or near-circular orbits also carries over from Newtonian physics to the relationship between the time dilation and the SR gamma. However, the GR correction to the perihelion precession of Mercury provides a sensitive distinction between GR and other competing relativistic theories of gravity. Basically, it comes down to a number, say 3, which can be fudged in many different ways if you don't have a fully resolved theory, but which is totally fixed by GR. (It is of course possible for other theories to give the same result too, and some of them do). If you want to look at something that looks like energy-momentum density in GR, study the Landau-Lifgarbagez pseudotensor. If you manage to work out how it maps back to Newtonian approximations (or find a text book which covers this), please let me know.
AWESOME! For a quick search this is exactly what I was looking for, but didn't know where to find! So basically this is an adjustment to the stress-energy tensor that allows energy and momentum to be conserved (which GR does not) by placing it in the gravitational field? How is this not more well known? I found a book on introductory book on Reimann Geomerty that I plan to read next week (over spring break). From the surface treatment I have read it appears that the Landau-Lifgarbagex pseutotensor is a function of the metric and the stress-energy tensor already used in GR. Am I correct to then think that using the Landau-Lifgarbagez pseudotensor to represent a system would produce the same results as GR?
No, this is incorrect. Local conservation of mass-energy and momentum follows directly from the Einstein field equations: http://en.wikipedia.org/wiki/Einstein_field_equations#Conservation_of_energy_and_momentum There is no adjustment needed in order to make conservation work. If I'm understanding the WP article http://en.wikipedia.org/wiki/Landau-Lifgarbagez_pseudotensor correctly, the only advantage of the L-L pseudotensor is that it allows you to make the conservation law nonlocal in some sense. The reason I say "in some sense" is that there are fundamental reasons why you can never have global conservation laws in GR. MTW has a good discussion of this at p. 457. Use of the L-L pseudotensor is not different from use of GR. The L-L pseudotensor is just a particular quantity that can be defined *within* GR. Again, you're really shooting yourself in the foot by not reading a standard textbook treatment of the subject.