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stub for Einstein-Hilbert action
added more quotes from Hilbert’s lecture at Einstein-Hilbert action – History
Some of this stuff tends to confuse me (speaking as someone ignorant of physics), so maybe Urs or someone else can help.
My basic question is: what is the parameter along which gravity is evolving? Usually I think of “dynamics” as being given by a map $t \mapsto S(t)$ where we think of “$t$” as a continuous or discrete “time” parameter, and $S(t)$ is a “state” (or something) of a system at time $t$. Dynamics thus conceived is evolution of a system along time. But here, we seem to be speaking of the dynamics of spacetime $(X, g)$ where “time” is already something (that is measured by local observers) in $X$. So I never understand what dynamics of spacetime means: what is the parameter along which spacetime is evolving?
This lack of understanding would extend to a (presumably minor) quibble about the way the Einstein-action is expressed in the article. We start off being given a pseudo-Riemannian manifold $(X, g)$. But then we have this action $(X, g) \mapsto some\; integral$ where $(X, g)$ is not given in advance but is presumably allowed to vary.
Isn’t the idea that somehow a particular metric $g$ is being chosen out of the space of possible metrics? So it’s more like the way a chain chooses to pick out a catenary from the space of possible curves in which it might hang. Except, of course, in the latter case we can at least envisage the other possible ways the chain can hang and think of the chain moving to adopt the catenary, whereas there seems no similar sense to the variation within the space of metrics.
David, when you say $g$ is “chosen”, do you mean chosen according to a least (Einstein-Hilbert) action principle?
Thanks for the questions. I have edited the entry slightly to clarify more.
The (vacuum) Einstein-Hilbert action is defined for a fixed (compact) manifold (with boundary) $X$ to be the function
$S_{EH} \;\colon\; Met(X) \longrightarrow \mathbb{R}$from the space of smooth (pseudo-)Riemannian metrics on $X$ to the real numbers, given by the assignment $g \mapsto \int_X R(g) vol(g)$.
(More properly the action is that function descended to the moduli stack $Met(X)//Diff(X)$, but I suppose this is not the subtlety that we are discussing right now.)
A solution to the (vacuum) Einstein’s equations of motion is a critical point of this function, hence is some (pseudo-)Riemannian metric $g$ on $X$.
In general this cannot be expressed as an “evolution along a parameter” and this state of affairs is part of what the word “general relativity” refers to.
If however we do assume a space/time split $X = [0,1] \times \Sigma_{d-1}$, then we can express solutions $g$ in adapted coordinates on $[0,1]$ and then they can be made to look like evolving along $[0,1]$. But one should remember that this involves arbitrary and non-intrinsic choices.
Thanks for your response, Urs. So if I read you correctly, the word “dynamics”, to have meaning, relies on a choice of “simultaneity slice” $\Sigma_{d-1}$ (or more properly, a decomposition $X \cong T \times \Sigma_{d-1}$) where, if initial data of $g$ and its time derivative restricted to this slice are given, then $g$ “evolves” along further slices $T = c$ according to the Einstein-Hilbert equations of motion obtained by a variational principle applied to the EH action.
given by the assignment $g \mapsto \int_X R(g) vol(g)$.
Would there be any objection to writing this, instead of $(X, g) \mapsto \int_X R(g) vol(g)$ as in the article? So that the compact manifold $X$ is understood as fixed from the outset?
But how should we think of a critical point for an action emerging in the first place in this case? Has anyone thought of the analogy with the path integral explanation for classical paths that quantum contributions sum around these, but cancel elsewhere? Could one have a quantum sum of metrics?
Going back a stage, why even was there a fixed spacetime to begin with? Do people imagine the whole ’space’ of possible spacetimes, and selection over that?
The crucial thing about the EH action seems to be put under the carpet, even not mentioned. I mean, the general covariance fact that Ricci scalar is scalar not only in the sense of changes of local coordinates (trivial), but also under the diffeomorphism group (nontrivial), This is a calculation of several pages. do not know if there is some hi level point of view which can make it more clear or more general. I could contribute to the page in this sense, but the only calculations I know are brute force old fashioned physics calculations; I would like to know first what Urs and other think about this issue.
Todd, sure, I have removed some of the $X$s in the domain, if that reads better. (But of course the functional is also defined on the space consisting of all manifolds with all metrics on them).
David, yes, the idea is that the critical points of the EH action are to be thought of as lowest order approximations to the path integral over that action as for any other action functional, too. Proposals to do quantum gravity by explicitly computing that path integral in some situations include Hawking’s old “Euclidean quantum gravity” and more recently Loll et al’s “causal dynamical triangulation” approach.
Zoran, you should always feel free to add what you feel is interesting to add. That’s the default intention, that each page should collect interesting information.
I have now added a pointer to section 4.2 in these MIT lecture notes which briefly discuss the diffeo invariance.
Urs, I made some minor edits in the history section of Einstein-Hilbert action to make the English sound more idiomatic to me – hope you don’t mind.
However, I wasn’t sure how to fix the grammar here:
This way notably at some point the supergravity variant of Einstein-gravity in the context of Riemannian supergeometry was found, a theory developing which eventually leads to superstring theory.
Okay, I have fixed it.
Zoran, you should always feel free to add what you feel is interesting to add.
Yes, but I still first need an answer to my question. Is it indeed nontrivial thing with several page of derivation, as it is in classical physics textbooks, or at some hi level of math there is a slick short way to get the fact that Ricci scalar is in fact not only tangent scalar but scalar under the whole diffeomorphism group ?
I wonder if a fully HoTT formulated physics could be easier to learn. Should any term that appears in texts like the one mentioned in #10 have its HoTT equivalent?
Let’s take the concept of a Killing vector field. So these are infinitesimal isometries. Now where do they feature in our configuration space account
$\ast : \mathbf{B} Diff(\Sigma) \vdash [\Sigma, \mathbf{Met}]?$My natural reaction is to reach for the external description $[\Sigma, \mathbf{Met}]//Diff (\Sigma)$. Then an isometry is a kind of diffeomorphism preserving each $(\Sigma, g)$. Will there be a clever dependent product type account?
Could you imagine thinking natively in the type theory? Perhaps people already do.
Zoran, I have nothing specific in mind, this is standard stuff. The lecture notes linked to above for instance give a short and standard proof of invariance under small diffeos. That would be the first thing one should put into the entry.
David, given a metric $g$, its Isometry group is the stabilizer subgroup of $g in Met(X)$ under the $Diff(X)$-action, hence is the loop space object
$Iso(g) \simeq \Omega_g (Met(X)//Diff(X)) \,.$The Killing vector fields are the Lie algebra to that group.
or better in HoTT language: the delooping of the Isometry group is the 1-image of the name of $g$
$\mathbf{B} Iso(g) = im_1( \ast \stackrel{\vdash g}{\longrightarrow} Met(X)//Diff(X) ) \,.$(sorry for the telegraphic style of the messages, don’t have much time today…)
Zoran, I have nothing specific in mind, this is standard stuff.
Thanks. I do not know what is standard and trust the books as they are, sometimes being wrong.
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