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Re: [Phys-L] GR and Gravitons



Regarding Jeffrey S's question:

I have a question about general relativity and the hypothetical
graviton. It is my understanding that the presence of a massive
object alters the geometry of spacetime thus causing the
observed motion of free particles in the vicinity of the massive
object. The particles aren’t experiencing a force but simply
following the contours of spacetime.

Yes.

The question is, why would anyone think that there would be an
exchange particle governing the (non-existent) interaction
between the massive object and any one of the particles.

But there *is* an interaction. It is just that the interaction is mediated by the geometry of spacetime. In the words of John Wheeler, "Spacetime tells matter how to move; matter tells spacetime how to curve." In other words the massive object curves the spacetime geometry in its vicinity in some appropriate manner, and the particles with nonzero mass follow the timelike geodesics on that curved spacetime (and the massless particles follow null geodesics). Thus the massive object communicates with the particles via the spacetime geometry intermediary.

The answer I have come across is that general relativity is a
field theory and fields can be quantized—the graviton is the
quantum of the field.

That's the story (assuming gravitation really *is* compatible with quantum field theory, and the classical limit of the mediating quantum field is classical GR).

Then the question becomes, what is the field? Is it spacetime
itself?

Essentially, yes. In a little more detail, the potential field (analog of the EM 4-potential, A) is the spacetime metric
tensor g. Supposedly a quantum version of gravity has the components of the g-tensor potential quantized in terms of creation & destruction field operators obeying appropriate bosonic commutation relations. The derivatives of the metric field operators are the Levi-Civita connection fields whose Newtonian & classical limit shows up as the Newtonian gravitational acceleration field g. (This Newtonian acceleration g is not to be confused with the metric tensor g above which unfortunately has the same basic symbol, even if though the components are labeled differently. BTW, the determinant of the metric tensor is also called g as well, so one has to be careful in understanding the meanings of the symbols used in gravitational theory.)

The ostensible format of the theory appears to break the diffeomorphism invariance (the particular gauge symmetry of GR) of the theory when it is quantized in a particular coordinate frame. But that is only a superficial breakdown. This is like when EM is quantized as QED in, say, the transverse Coulomb gauge; it also appears to break the local gauge and Lorentz invariance, but that is also a superficial breakdown.

One big difference between GR & EM is that the physical fields in GR are the 2nd derivatives of the potential fields, rather than the first derivatives, as they are in EM. In EM the A-potential components don't have a physical meaning themselves, rather the E & B field, which are the first derivatives of the A-potential components, are the physical force fields acting on the charged sources and whose values are constrained by the requirements of Maxwell's equations. But in GR neither the metric tensor components nor the (first derivative) Levi-Civita connection components are the physical fields (despite the later giving the Newtonian acceleration field). The components of the curvature tensor involve derivatives of the Levi-Civita connection, i.e. 2nd derivatives of the metric tensor, and it's the curvature tensor's components are the physical fields in GR whose components are constrained by the requirements of Einstein's field equations, and they show up as the gradient of the Newtonian acceleration field as the causes of tidal effects in the Newtonian theory.

This page http://www.quantum-field-theory.net/einstein-didnt-say/
has another suggestion. Is that page a bunch of nonsense?

After a very cursory glance at the page it only seems to advocate a change in traditional emphasis at the introductory level for GR, rather than saying anything manifestly wrong or nonsensical.

Dave Bowman