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Excerpt from the textbook below. It seems ambiguous what the author means and I am unable to proceed.

Imagine that you live in a Universe where Einstein never existed. Instead, he was replaced by someone that looks alike called Feinstein. Feinstein never envisioned the idea that time and space are on equal footing (something like Cartesian geometry).

Now imagine that you are Feinstein, and you want to describe gravity as geometry, as is done in General Relativity, but without relative time.

Then you would want to describe gravity as curvature in a 3-dimensional spatial manifold. Now imagine that you are in a region where the Feinstein equations hold in three spatial dimensions and that you are in vacuum so that the stress tensor is zero. The question is, would someone on the surface of a planet feel gravitational attraction to the planet if gravity were described as curvature in three dimensions?

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    $\begingroup$ A possibly relevant reference: Jackwiw & 't Hooft Three-dimensional Einstein gravity: Dynamics of flat space sciencedirect.com/science/article/pii/… $\endgroup$ – Anders Sandberg Apr 8 '18 at 5:34
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    $\begingroup$ Unclear what you are asking. No time component to what? Ricci tensor? Stress-energy tensor? Metric? 3D euclidean gravity seems to be also not what you are asking about. Are you interested in ultrastatic space-times? $\endgroup$ – A.V.S. Apr 8 '18 at 7:10
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    $\begingroup$ I don't think this is possible to answer. You're asking about a 3D Riemannian manifold, and the geodesic equation will still apply but the variable used to parameterise the geodesics, $\tau$, would just be the proper distance along the geodesic and not a time. You would still find initially parallel geodesics converge or diverge as $\tau$ changes, but without a time dimension I can't see how you could describe this as an acceleration. The question would make more sense if you retained a time dimension but required the signature be positive definite. $\endgroup$ – John Rennie Apr 8 '18 at 11:36
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    $\begingroup$ If you use a 3-dimensional Einstein equation in 3d spatial time, where do you get time dependence in the theory? Gravitational waves? Also, Riemann tried to construct a theory of gravity as curvature, and couldn't because he didn't know about the Minkowski metric signature. $\endgroup$ – Jerry Schirmer Apr 9 '18 at 4:03
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    $\begingroup$ Excerpt from the textbook below. What textbook? Please don't cut and paste random stuff without attribution. $\endgroup$ – Ben Crowell Apr 9 '18 at 20:37
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I agree, there is ambiguity in specifying what the author (of your text) meant. By saying but without relative time the author seems to imply that there is absolute time, but does this mean that there is still finite speed of light and relativistic equations of motions for particles or is there a Galilean mechanics in a curved space? At the very least, existence of time variable is implied by the presence of sentient being and by phrases like feeling the attraction.

So, we have spatial curvature described by metric $g_{ab}$ that should satisfy some sort of 'Feinstein equations'. If we assume that those are $$ R_{ab} =8\pi G (T_{ab} −Tg_{ab}),\tag{*} $$ then outside of material bodies ($T_{ab}=0$) we would have $R_{ab}=0$. But in three dimensions Ricci tensor fully defines Riemann curvature tensor since Weyl tensor vanishes identically in 3D. So outside of bodies we have $R_{abcd}=0$, i.e. flat space. Test particle moving outside of material body would be moving along a straight line and so would not feel gravitational attraction. However, this does not mean that such space-time would be trivial. Inside the material bodies the space would be curved, which means that around a body there would be a solid angle deficit (or excess). For example a space outside of a single spherically symmetric body would have a metric $$ ds^2=a \cdot dr^2 +r^2(d\theta^2+\sin^2\theta d\phi^2), $$ with $a>1$ (angle deficit) for positive curvature inside the body or $a<1$ for negative curvature and angle excess. So a system of point-like particles would have a geometry of a 3-polytope (potentially with nontrivial topology) with particles at its vertices.

Presumably, that is all that the textbook would expect from this exercise. However, I am not at all satisfied with 'Feinstein equations' (*). First, they do not have any dynamics outside of possible time dependence of $T_{ab}$, while I would expect equations to include time derivatives of $g_{ab}$. Second, if we only consider spatial curvature, there is no reason not to include derivatives of Ricci tensor, $\nabla_a R_{bc}$ and higher, in equations (in ordinary GR they are excluded since we do not want higher than second time derivatives of metric), so there could be other possible equations as a replacement for GR.

And so there is another interesting model for a general relativity with space and time on different footing, this one with a view toward quantum gravity: Hořava–Lifshitz gravity. This model is written in terms of spatial metric, lapse function and shift vector of the ADM formalism. However, instead of starting with Einstein-Hilbert action, the model achieves power-counting renormalizability by using different scaling for space and time. The resulting theory is explained in original paper:

  • Hořava, P. (2009). Quantum gravity at a Lifshitz point. Physical Review D, 79(8), 084008, doi, arXiv.

Abstract:

We present a candidate quantum field theory of gravity with dynamical critical exponent equal to z=3 in the UV. (As in condensed-matter systems, z measures the degree of anisotropy between space and time.) This theory, which at short distances describes interacting nonrelativistic gravitons, is power-counting renormalizable in 3+1 dimensions. When restricted to satisfy the condition of detailed balance, this theory is intimately related to topologically massive gravity in three dimensions, and the geometry of the Cotton tensor. At long distances, this theory flows naturally to the relativistic value z=1, and could therefore serve as a possible candidate for a UV completion of Einstein’s general relativity or an infrared modification thereof. The effective speed of light, the Newton constant and the cosmological constant all emerge from relevant deformations of the deeply nonrelativistic z=3 theory at short distances.

or a recent review:

  • Wang, A. (2017). Hořava gravity at a Lifshitz point: a progress report. International Journal of Modern Physics D, 26(07), 1730014, doi, arXiv.
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