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I don't understand your objection to the slicing procedure. The central assumption is that for a reasonably physical spacetime there exists a unique maximal gl

I don't understand your objection to the slicing procedure. The central assumption is that for a reasonably physical spacetime there exists a unique maximal globally hyperbolic solution to the field equations, justifying the use of the initial value formulation of General Relativity. [aside: [1]] Once we do that we can take any arbitrary spacelike hypersurface Σ and in ADM-ish vocabulary observe the compact closure conditions (Σ has a spacelike geodesic from every p in Σ to every other p in Σ; every maximal spacelike geodesic has a nonempty open segment in Σ; every maximal timelike geodesic intersects Σ), the "thin" condition (no nomepty segment of a null geodesic is found in Σ), and a variety of causal conditions (e.g. no timelike geodesic connects points in Σ; no maximal timelike or null geodesic intersects Σ more than once). One can translate these features to your cones-only view. On a Minkowski diagram with p at the origin, Σ can have any shape or orientation compared to the light cones as long as p -- and nothing else in either of p's light cones -- appears on Σ. We can do the same for any other p in Σ, or we can decide whether the origin of any pair of light-cones is in Σ or not. We can also probably build Σ by hand from a principled choice of available light-cones. (One would be tempted to call that "initial data"). Is Σ physical? Of course not! But it can be used as the basis for physics that describes point-coincidences on other spacelike hypersurfaces, which we can then verify. > The future light cone is what can possibly be changed based on things we do here and now In which of your three regions do we find the freedom to do this? Is here-and-now not fully determined by the "fixed and certain" content of its past lightcone? Is the "we" part of "what we do here and now" describing a point on the manifold or is it describing a set of points on the manifold? Choose a point somewhere in the middle of "we" -- in which of the three regions are its other points? I struggle to answer that without thinking about an extended "we" with many entries in Σ (including photons (and other carriers of body forces) bouncing around within or even passing through the extended body). Maybe it's much easier for you. I wonder how you'd do if you were to think of grossly non-Eulerian (and even non-timelike) observers in a Big Crunch universe. In a Big Bang universe all points are in the future light cone at BB; in a BC universe all points are in the past light cone at BC. At every other point, only some points are in the two lightcones at that point. We get a BC from a time-inversion of our BB->approx de Sitter universe (e.g. for an observer with differently-aligned arrow-of-time). Assuming a long period of matter sparseness, in a BB->BC universe, we also get a ~BC universe if we cut away the expanding part; if we cut away the recollapsing part, we get something that's hard to distinguish from a BB->~dS universe like our own. In a BB->~dS universe I think we'd agree that the future light cones of two observers coincident at p can eventually fully separate. Can we agree that in a BC universe the past light cones of two observers at p can be at least somewhat, or even (practically) fully, non-overlapping (our observers need not be either "Eulerian" comovers with the metric contraction, and can even move lightlike, and there can be a cosmic deflation just prior to the big crunch)? In an ADM(-like) picture we can fairly straightforwardly use ancestor slices of Σ in the foliation and decide that a pair of photons will be at some point p on Σ. I also think it makes a slice through a spacetime-filling electromagnetic field easier to think about, ditching the idea of bare photons as independent objects. Again it doesn't matter about the timelike direction of the slicing, and one might want to recalculate slicing along some other choice of timelike axis to try to spot coordinate artifacts. However maybe you might object that if we don't have full data for the ancestor of Σ, we're somehow not better off than in the final sentence of your second paragraph. - -- [1] Weaker causality, such as stable causality in the sense that arbitrarily small metric perturbations don't give rise to CTCs, can still be used but at the cost of solution uniqueness, and weaker still at the cost of smearing out the Cauchy surface so that initial data is not all at t=0 but in some function defined over 0 >= t >= -V^{-1} where V is the scale of strong causality violation.

pdonis 4y ago
> I don't understand your objection to the slicing procedure. It's simple: any "slicing" is frame dependent. And relativity says that things that are frame dependent do not have any physical meaning. > The central assumption is that for a reasonably physical spacetime there exists a unique maximal globally hyperbolic solution to the field equations Yes, but globally hyperbolic spacetimes still do not have unique slicings. There are still an infinite number of possible slicings, corresponding to the infinite number of possible inertial frames in standard SR. > In which of your three regions do we find the freedom to do this? If the universe is fully deterministic, then in one sense of course there is no "freedom" to change anything. But we don't know that the universe is fully deterministic (and QM strongly suggests that it isn't). But whether it is or not, the distinction I described is still there and still valid, since that distinction is about what we know at a particular event in spacetime. > In a Big Bang universe all points are in the future light cone at BB No, they aren't. The Big Bang is not a single point. It's a spacelike line. There are an infinite number of causally disconnected points one "instant" after the Big Bang. Similar remarks apply in time reversed form to the Big Crunch in spacetimes that have it.
- raattgift 4y ago
  I agree with all your points except: > QM strongly suggests that it isn't an argument over which is certainly out of scope here (however it did answer my question); and > The Big Bang is not a single point. It's a spacelike line which I think is wrong for the standard cosmology. See Davis & Lineweaver 2004 <https://ui.adsabs.harvard.edu/abs/2004PASA...21...97D/abstract https://ui.adsabs.harvard.edu/abs/2004PASA...21...97D/abstra...> notably fig. 1 and the expalanatory text scattered within. Why would (cosmological) t=0 be a line rather than a point, a 3-volume, or some even weirder non-Lorentzian structure? I can't think of why it would be a line other than (reaching here) nonvanishing vorticity (in either the sense of the Raychaudhuri equation or because of ~axisymmetry with spin), and those aren't features of the standard cosmology. Obviously I've seen Penrose diagrams with the BB drawn out to look like a horizontal line, but have never taken that to be anything other than for legibility or print-reproducibility reasons, similar to the dimensional reduction of the diagram itself. I would be grateful for a reference with explanatory text if you think this is wrong. Absent published authority, I think you could also take this up with Pössel (Max-Planck-Inst. Astro. Haus der Astronomie) who by sheer coincidence ran a related twitter thread <https://twitter.com/mpoessel/status/1551119783315251200 https://twitter.com/mpoessel/status/1551119783315251200> in which he writes (among other things): " ... 23/ ... Let's forget about the hot Big Bang phase, pretend that 24/ all the galaxies taking part in cosmic expansion are point particles corresponding to galaxies we see in the observable universe today were at one and the same location. 25/". @johncarlosbaez has been all over that thread on Twitter too, and would probably engage with your it's-not-a-point-it's-a-line point. I am pretty certain that Pössel means we ignore the "hot Big Bang phase" to mean not just the first few minutes (up to the synthesis of helium) (which is a fairly common time frame for "hot Big Bang") but up to early structure formation, and that he'd clarify if asked. I agree that in the standard cosmology there are an infinite number of causally disconnected points one "instant" after the Big Bang, but don't see how that can't follow from the standard BB singularity. Of course a BC universe is not standard cosmology, so alternative final shapes don't raise my eyebrows at all, unless you wish to argue that in General Relativity, no BC universe can even in principle have a point wherein all other points are in its past light cone. All that said, I'm not sure why you're objecting to slicing as having a dependency on choice of coordinates. I did write that \Sigma is not physical. I will now write that it is very convenient to calculate in some set of coordinates, and that it is possible not to fall into holes (pardon the pun) while doing so. Finally, I'd like to turn on its head your second sentence. Relativity allows some physical distributions of stress-energy to pick out a meaningful set of coordinates. (as has been known in cosmology for many years. cf. <https://arxiv.org/abs/1211.6338 https://arxiv.org/abs/1211.6338> §2 notably p 4)

pdonis 4y ago

> I don't understand your objection to the slicing procedure. It's simple: any "slicing" is frame dependent. And relativity says that things that are frame dependent do not have any physical meaning. > The central assumption is that for a reasonably physical spacetime there exists a unique maximal globally hyperbolic solution to the field equations Yes, but globally hyperbolic spacetimes still do not have unique slicings. There are still an infinite number of possible slicings, corresponding to the infinite number of possible inertial frames in standard SR. > In which of your three regions do we find the freedom to do this? If the universe is fully deterministic, then in one sense of course there is no "freedom" to change anything. But we don't know that the universe is fully deterministic (and QM strongly suggests that it isn't). But whether it is or not, the distinction I described is still there and still valid, since that distinction is about what we know at a particular event in spacetime. > In a Big Bang universe all points are in the future light cone at BB No, they aren't. The Big Bang is not a single point. It's a spacelike line. There are an infinite number of causally disconnected points one "instant" after the Big Bang. Similar remarks apply in time reversed form to the Big Crunch in spacetimes that have it.

raattgift 4y ago
I agree with all your points except: > QM strongly suggests that it isn't an argument over which is certainly out of scope here (however it did answer my question); and > The Big Bang is not a single point. It's a spacelike line which I think is wrong for the standard cosmology. See Davis & Lineweaver 2004 <https://ui.adsabs.harvard.edu/abs/2004PASA...21...97D/abstract https://ui.adsabs.harvard.edu/abs/2004PASA...21...97D/abstra...> notably fig. 1 and the expalanatory text scattered within. Why would (cosmological) t=0 be a line rather than a point, a 3-volume, or some even weirder non-Lorentzian structure? I can't think of why it would be a line other than (reaching here) nonvanishing vorticity (in either the sense of the Raychaudhuri equation or because of ~axisymmetry with spin), and those aren't features of the standard cosmology. Obviously I've seen Penrose diagrams with the BB drawn out to look like a horizontal line, but have never taken that to be anything other than for legibility or print-reproducibility reasons, similar to the dimensional reduction of the diagram itself. I would be grateful for a reference with explanatory text if you think this is wrong. Absent published authority, I think you could also take this up with Pössel (Max-Planck-Inst. Astro. Haus der Astronomie) who by sheer coincidence ran a related twitter thread <https://twitter.com/mpoessel/status/1551119783315251200 https://twitter.com/mpoessel/status/1551119783315251200> in which he writes (among other things): " ... 23/ ... Let's forget about the hot Big Bang phase, pretend that 24/ all the galaxies taking part in cosmic expansion are point particles corresponding to galaxies we see in the observable universe today were at one and the same location. 25/". @johncarlosbaez has been all over that thread on Twitter too, and would probably engage with your it's-not-a-point-it's-a-line point. I am pretty certain that Pössel means we ignore the "hot Big Bang phase" to mean not just the first few minutes (up to the synthesis of helium) (which is a fairly common time frame for "hot Big Bang") but up to early structure formation, and that he'd clarify if asked. I agree that in the standard cosmology there are an infinite number of causally disconnected points one "instant" after the Big Bang, but don't see how that can't follow from the standard BB singularity. Of course a BC universe is not standard cosmology, so alternative final shapes don't raise my eyebrows at all, unless you wish to argue that in General Relativity, no BC universe can even in principle have a point wherein all other points are in its past light cone. All that said, I'm not sure why you're objecting to slicing as having a dependency on choice of coordinates. I did write that \Sigma is not physical. I will now write that it is very convenient to calculate in some set of coordinates, and that it is possible not to fall into holes (pardon the pun) while doing so. Finally, I'd like to turn on its head your second sentence. Relativity allows some physical distributions of stress-energy to pick out a meaningful set of coordinates. (as has been known in cosmology for many years. cf. <https://arxiv.org/abs/1211.6338 https://arxiv.org/abs/1211.6338> §2 notably p 4)