Markus Hanke

It’s possible to consider discrete models of fluid dynamics, but generally speaking these don’t take the form of neat, closed formulas that can be easily worked on paper - rather, you’ll be dealing with numerical models on powerful computers. An example of this is CFD-DEM. The standard Navier-Stokes equations are then just the continuum limit of such a model.

Are you familiar with the topological arguments put forward in MTW with regards to why the EFE has the form it does? Not sure if one can call it a “derivation” exactly, but it’s nonetheless very interesting.

Yes, but then you’d end up with a different theory of gravity. In GR, the tensor we’re looking for can only contain first and second derivatives of the metric, must be linear in the latter, symmetric, and divergence-free. Lovelock’s theorem guarantees that the only tensor that fulfills these criteria is the Einstein tensor.

It’s best to look at the Einstein tensor as a kind of machine with two slots - if you feed into both slots a unit vector that points into the future time direction, it will give you as result a real number that represents the average Gaussian curvature in a small spatial region around the event you are working at: \[G_{\mu\nu}t^{\mu}t^{\nu}=k\] So the Einstein tensor is a kind of average curvature around some event. I don’t think it is helpful to try and find specific “meanings” for specific components.

Are you sure this is right? Admittedly I don’t know so much about the history of Christianity, but I’m pretty sure there weren’t any churches at the time these texts were written…

! Moderator Note Moved to Speculations, as this is completely at odds with mainstream astrophysics.

It’s called that because the energy-momentum tensor that corresponds to this metric (FLRW) is actually one describing a perfect fluid, with its constituents being the galaxies, clusters etc that make up the cosmos. So FLRW is basically an interior fluid solution on a very large scale. Indeed. This coordinate system is called Gaussian normal coordinates.

While this is of course true, I think it’s very important to remember that such a transformation changes the physical meaning of the time coordinate - it will no longer correspond to a clock co-moving with the cosmological fluid.

Lol…nice analogy 😉

Would it not correspond to a clock that is in motion relative to the cosmological fluid?

Yes, such a transformation would just be a diffeomorphism, which is of course always allowed. The disadvantage though is that it changes the physical meaning of the time coordinate - it then no longer corresponds to a clock co-moving with the cosmological fluid. I also suspect (not sure though) that this would introduce off-diagonal terms into the metric? I think everything considered, the usual Gaussian normal coordinates probably give the simplest and easiest to understand form of the metric.

Measurements of space and time at different places/times are related via the metric, which is the basic dynamic variable in GR. This is the entire point of the model, so you don’t need to introduce anything new. The difference to SR is that frames are no longer related via simple Lorentz transformations, so time dilation and changes in lengths do not necessarily carry the same factors, nor do they even necessarily occur together. No. Spacetime curvature behaves quite differently from elasticity in a medium; the image is just an analogy and visualisation aid.

I don’t know what ”proportional relativistic variable” is supposed to mean, but we know that measurements of distance are observer-dependent in GR, as several people here have already pointed out. So again - where is the “problem”? No, it’s not a contradiction, because in GR there’s a big difference between proper quantities and coordinate quantities. The two observers use different local notions of time, so you’re not comparing like for like. You compare spatial distances just like you do time separations, and you find that they are observer-dependent, as expected. Where is the “problem”?

In curved spacetimes, measurements of spatial distance, just like measurements of time, generally depend on the observer, because those are not tensorial quantities. Only spacetime intervals are. This is well understood, and the mathematics of differential geometry on Riemann manifolds predate GR. Exactly where is the problem with this?

I don’t actually see what the supposed “problem”, that the title of this threads alludes to, is meant to be. In curved spacetimes, observers will, in general, not agree on quantities that aren’t either tensors, or invariants formed from them. Therefore we do not, in general, expect observers to agree on specific measurements of space, time, energy, momentum etc - but they will always agree on the metric, and thus spacetime intervals and everything that derives from this. That’s just basic differential geometry. So could you just simply state what you think the “problem” is with this, preferably without getting lost in the intricacies of some specific scenario?

It doesn’t assume this, it’s how the maths work out. The FLRW metric is a solution to the Einstein equations - so you start by putting in your initial and boundary conditions, and solve the equations; the result is FLRW. The initial assumptions are homogeneity and isotropy, which taken together already constrain the final metric enough to give it its general form. To obtain the exact form, you begin with the standard energy-momentum tensor for a perfect fluid, feed it into the field equations, and solve. The result is that all coordinate dependencies drop out, except the time dependence of the spatial part. FLRW is a Petrov-type O spacetime; metric expansion (meaning: measurements of distance depend on when they are performed) is a general geometric property of many such spacetimes, and by no means exclusive to FLRW. Yes, Lambda-CDM doesn’t provide a perfect fit to all data, but it provides the best fit amongst all currently known cosmological models. But I agree that it probably won’t be the last word on the subject. There is of course always the possibility that new physics exist which we are not yet aware of. If so, our cosmological models will need to change. FLRW provides the best fit to the data, based on the physics (ie GR and fluid dynamics) which we currently know. Again, you can try and find a solution to the field equations that has a different form, yet still fits the available observational data. I would be interested in seeing it. Again, the FLRW metric is a solution to the field equations of GR, it has not just been invented “by hand” - so the best answer to your question is that the laws of gravity determine it to be so. So far as I can see (having worked through this entire solution process myself), every time you start with homogeneity and isotropy as basic symmetries, you’ll end up with a metric of the general form of FLRW, and the precise spatial coordinate functions will depend on the energy-momentum tensor you use in the field equations. For this to come out different, you’d have to amend the laws of gravity itself.

Local relative motion in a static background would be limited to subluminal speeds in accordance with the usual laws of kinematic, so for redshift we’d find z<1 always, whereas with metric expansion there is no such limit. Furthermore, if there is only local motion in an otherwise static space, then some of these objects will recede from one another, whereas others approach each other, like molecules in a gas. We’d see a mix of both blue- and red-shift, unless you want to postulate that we are the Center of the universe, and everything moves radially away from us for some reason, which is not very plausible. But with metric expansion, it’s the space in between that “expands” (I don’t like this term, but it has become standard), so on the largest scales everything will appear to recede from everything else, and it will do so the same way no matter what direction you look at, and irrespective what’s in between here and there. Also - if the rate of apparent recession isn’t constant (which is what seems to be the case), then, if you were to deal with local motion, you would have to have either some mechanism of acceleration, or some explanation as to why everything falls away from us. Overall you’d end up with a model that’s actually much more complicated and much less plausible than metric expansion.

The cause of redshift is an increase in distances between objects that are not gravitationally bound to each other; it’s not motion in the classical sense, but what’s called apparent motion. This is different from the claim that the objects somehow accelerate to greater and greater (superluminal!) speeds - that’s evidently not what happens. To be more precise, metric expansion means that the outcome of distance measurements on the same set of points depends on when the measurements are performed - as you age into the future, the measured distance between any two points will increase, unless they are gravitationally bound. Thus, all points will appear to recede from all other points - which, again, is different from all points moving radially outwards from a common center.

Yes, definitely. But I think I was actually wrong in my comment - upon closer inspection, it really does seem to rule out MOND altogether. Subject to further verification, of course.

As it should be, of course This doesn’t actually rule out MOND, it just means that even if MOND holds true, you still need dark matter to fit all observational data, though presumably in lesser quantities.

Because it is based on the FLRW solution to the EFE, and therein only the spatial part of the metric is non-trivial and carries an expansion factor, there is no time dilation in this cosmological spacetime. Both GR in general and its particular solution for this case, the FLRW metric, predate Hubble’s observational findings. Metric expansion is a direct consequence of the laws of gravity, for any homogenous and isotropic distribution of energy-momentum that meets certain criteria; it’s not an independent, stand-alone idea. Accepted as the basis for a model in the context of cosmology, yes, but not invented or theorised - see above. Redshift was historically the first observational evidence that became available to us, but nowadays the Lambda-CDM model covers many other observations too, which were made after Hubble, eg the CMBR with its polarisation, the Ly-wavelength g-wave background, large-scale structure, acoustic baryonic oscillations, ratios of light/heavy elements etc. Among all proposed cosmological models, it is the one that fits the body of all available data the best - though it almost certainly won’t be the last word, I dare predict, because it also does have its problems. You are welcome to try and find a solution (that isn’t just a trivial diffeomorphism) to the EFE that leads to such a law. Remember that it should also be compatible with all other observations, in order to be a useful cosmological model.

Looks like the air might be getting a bit thin for MOND: https://academic.oup.com/mnras/advance-article/doi/10.1093/mnras/stad3393/7342478?login=false

Because it is directly correlated to the distance of the source object in question, and that relationship is the exact same no matter in which direction we look, and no matter what else is/is not between here and there. Also, objects don’t just recede from us, but also from each other. One must also remember that redshift is only one of several consequences of metric expansion; it’s by no means the only data point.

Current AI’s are language models, they possess no real understanding of the subject, and as such their answers are quite often wrong or misleading. That is why they are not valid sources of scientific information. So is Newtonian gravity - we measure forces because there is gravity, and there’s gravity because a force is exerted. So what? The model still works well within its specific domain, which is why we still learn it at school. But go outside of its domain, and it fails miserably. You yourself just quoted a paper that provides an example of some of these local effects being measured. But like I said, they are very small compared to the z~10 cosmological redshifts.

Redshift measurements are not done by observing anything “getting smaller”, so you don’t need to look for millions of years. Also, it’s not that things recede only from us - everything recedes from everything else. The distances between all systems not gravitationally bound will increase over time, equally in all directions. Well, it’s just that cosmological redshift is of a vastly greater magnitude than any local gravitational or kinematic effects - eg frequency shift due to the relative motion of galaxies is on the order z ~ 0.005 or so, whereas typical cosmological redshifts for very distant objects is of the order z ~ 10. That’s many orders of magnitude (due to the maths relations involved), so local effects such as the ones you mentioned can be safely ignored on very large scales.