Applicability of Spin-2 Field Interpretation of GR

[Matterwave]

GR has its own set of interpretations: spacetime curvature is one, "spin-2 field on a flat spacetime background" is another. (The latter is technically limited to spacetimes with topologies that can have a flat background put on them.) The second is kinda sorta LET-ish, I suppose (the flat background is in principle unobservable), but it has no historical lineage in common with LET that I'm aware of.

Sorry to push us more afield -- does the spin-2 field picture "work" in non-perturbative (strongly curved) spacetimes?

 

[PeterDonis]

Sorry to push us more afield -- does the spin-2 field picture "work" in non-perturbative (strongly curved) spacetimes?

Please give a specific example as "non-perturbative (strongly curved)" is too vague.

 

[Matterwave]

Please give a specific example as "non-perturbative (strongly curved)" is too vague.

Schwarzschild solution near the event horizon?

I don't know much about the spin-2 picture so just wanted to know if it genuinely reproduces GR or only in "weak field limit".

 

[PeterDonis]

Schwarzschild solution near the event horizon?

The spin-2 field model handles that just fine. Note that curvature near the event horizon is not necessarily "strong"--it goes like the inverse square of the Schwarzschild radius, so as the hole gets larger, the curvature at the horizon gets smaller.

I don't know much about the spin-2 picture so just wanted to know if it genuinely reproduces GR or only in "weak field limit".

It's not limited to that. As I said, it handles any solution whose global topology is compatible with a flat metric. (Note that the "weak field limit" of GR is not a single well-defined thing, because the term "field" can refer to different things.)

The maximal extension of the Schwarzschild solution has global topology $R^2 \times S^2$, which is not compatible with a flat metric (because of the $S^2$ part). But a patch of the Schwarzschild solution that is restricted to outside the horizon has topology $R^4$, which is obviously compatible with a flat metric.

 

[Matterwave]

The spin-2 field model handles that just fine. Note that curvature near the event horizon is not necessarily "strong"--it goes like the inverse square of the Schwarzschild radius, so as the hole gets larger, the curvature at the horizon gets smaller.


It's not limited to that. As I said, it handles any solution whose global topology is compatible with a flat metric. (Note that the "weak field limit" of GR is not a single well-defined thing, because the term "field" can refer to different things.)

The maximal extension of the Schwarzschild solution has global topology $R^2 \times S^2$, which is not compatible with a flat metric (because of the $S^2$ part). But a patch of the Schwarzschild solution that is restricted to outside the horizon has topology $R^4$, which is obviously compatible with a flat metric.

This is interesting to know! I always saw it, a long time ago, expressed starting with some perturbation assumptions like "let's express the metric as $$g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$$" and go on to analyze the h from there. But my memory is so vague it does not bear repeating any further lol.

 

[PeterDonis]

I always saw it, a long time ago, expressed starting with some perturbation assumptions

That was how the spin-2 field theory was first investigated in the 1960s. However, it took quite a few years to realize that the theory is not renormalizable and trying to assess convergence of the perturbation series by brute force doesn't work. In the late 1960s and early 1970s, Deser and others figured out a non-perturbative way of writing the theory, which made it clear that the final field equation is just the Einstein Field Equation of GR.