Markus Hanke

Is your density distribution a continuous function, or are you dealing with a collection of point masses? At first glance your system of equations looks ok, though I’m not immediately sure whether (3) is actually needed at all, as (1)+(2) should already imply (3) - I haven’t explicitly checked though. As for numerical methods, that’s not my area of expertise, so I can’t offer any suggestions; I’d say it would be very difficult to implement that in code without constraining the form of the density and velocity functions in some way. How will the user of the software input the functions? Will there be an interface where the user types the functions symbolically (in which case you need to implement a suitable parser), or do they go directly into the source code, or what did you have in mind?

There is no global law of energy conservation in curved spacetimes (of which FLRW is an example), so this is a non-sequitur. Energy-momentum is conserved only locally.

Geometrodynamics is an epistemological model of gravity, not an ontological one (IOW it’s a mathematical model) - we use it because it is amenable to the scientific method, and thereby found to work very well. One day in the future it will almost certainly be understood as an effective approximation to something more fundamental, which may then use an entirely different notion of time, or even dispense with the concept altogether. So Einstein didn’t offer any ontological explanation for time, he just took the notion at face value (“time is what clocks measure”) and used that to formulate a model of how gravity works. Quite successfully so, I might add. Hence, belief (in the religious sense) doesn’t come into this; rather, it’s about an epistemological description of some aspect of reality, and its usefulness to match experiment and observation.

It “warps” and “curves” (in the sense of geodesic deviation), but it doesn’t “twist” - in GR there is (by definition) no torsion.

How do you define “amount of space”, exactly? The FLRW metric is not a vacuum solution to the field equations (unless in the trivial case of a(t)=const.), so it doesn’t apply to a 2-body system in vacuum, such as the Earth-Moon system. I’d just like to stress - because I think that it is important to make this very clear - that this is an analogy, a way to look at the situation that can be used as a helpful conceptual aid under certain circumstances, just like the “rubber sheet” analogy. It is not to be taken literally, however, as spacetime isn’t a medium that “flows” somehow.

The MM experiment is only an isolated example of a very long history of extensive searches for an “aether”, and it has long since been superseded by far more sensitive experiments. No trace of any kind of aether has ever been found, even after 300+ years and many hundreds of experiment ranging from mechanical table-top setups right up to ultra-sensitive and complicated optical measurements of various kinds. On the other hand, local Lorentz invariance (the symmetry underlying Special Relativity) has been experimentally established to such a high degree that its validity is no longer in any kind of reasonable doubt. If you combine these, then the absence of any evidence for aether, as well as the simultaneous experimental verification of Lorentz invariance, make it - for all intents and purposes - a near certainty that there is no such thing as an aether. And why should there be? It isn’t needed in any way, shape or form to explain anything.

It’s mostly a matter of self-consistency. It is indeed possible to formulate field theories that retain local Lorentz invariance (i.e. they don’t outright violate SR), have real mass, and respect all conservation laws, yet allow for superluminal excitations. The trouble with these models is that they are not self-consistent, in the sense that they don’t have a well defined causal structure. To put it simply, the existence of tachyons (even if they don’t violate any other physics) would allow you to construct physical paradoxes that can’t be resolved in a self-consistent manner - which essentially rules out such models. This is one of those cases that are mathematically possible, but physically meaningless, so tachyons almost certainly do not exist.

So far as I can see there are roughly four basic approaches in the literature as to the nature of DM: 1. It is what it says on the tin - a form of matter that does not interact with light. This will likely require a new addition to the Standard Model, since none of the hitherto known particles readily appear to have the properties required of DM. This appears to be the most popular option that most scientists in this field pursue. 2. A new fundamental interaction. This postulates an as yet unknown additional fundamental interaction which acts on ordinary matter-energy. Hence, the motion of bodies we detect is the net result of both gravity and that new interaction 3. A modification of the laws of gravity. The idea here is that in actual fact there is no DM, but that gravity on larger scales is not well described by the GR field equations, requiring some amended (scale-dependent?) law of gravity on those scales. The appearance of DM is then simply the difference between what GR predicts, and what the actual motion of test particles under the amended gravity law is like. 4. DM is neither a new form of matter, nor a new interaction, nor the result of a new law of gravity. Rather, it arises because we are not using the standard (unmodified) laws of GR correctly. The idea here is that any calculation in GR relies on some form of simplified approximation that allows us to actually perform the computation - we must choose to ignore some boundary conditions, and introduce extra symmetries that in reality are not actually there, otherwise the equations are simply too complicated to be solvable, even in principle. For example, when modelling a galaxy, we might choose to describe it as a disk-shaped continuous gas distribution of roughly the right shape, which allows us to find some kind of solution to the field equations. In actual reality though a galaxy isn’t continuous like that, it’s a collection of a very large number of discrete objects that all interact gravitationally, so it’s really a general relativistic n-body problem with n being on the magnitude of ~100’s of millions. We assume that our continuous approximation to an actual galaxy yields a gravitational metric that is sufficiently similar to that of the (unsolvable) case of having 100 million discrete objects - but how do we actually know this, since we cannot derive an actual solution for the latter case? The Einstein equations are highly non-linear, so it is notoriously difficult to mathematically determine what kind of error arises from a given choice of simplification, and how this error evolves (kind of similar to varying initial conditions in chaos theory). So in this proposal, DM is precisely the error that arises from our choice of simplifications in the ansatz of our model - the idea being that if we were able to accurately model the gravitational source and the relevant boundary conditions, this error would simply disappear, and GR would produce the correct motion of all objects. So DM is an artefact of our own computational limits, and not a real aspect of the world at all. Out of all these, option (1) is probably the most popular and perhaps also the most likely, based on current knowledge. But we shall see.

On the contrary - time is integral to gravity as we observe it. There would be no gravity outside of massive bodies, if time did not exist - this can be formally shown. You just said that gravity and time exclude one another - but metric expansion is a gravitational effect. You seem to be contradicting yourself. Gut feeling is very useful in many contexts, but gravitational physics is not one of them.

It is the outcomes of measurements of time that are numbers, being the readings on idealised clocks. That is a subtle but important difference. Also, just because something can be quantified does not imply that it is an illusion - that’s a non-sequitur. The reason we know that time isn’t just an “illusion” is first and foremost the existence of gravity, specifically the tidal aspects of gravity. If you had only three spatial dimensions, but no time, it can be formally shown that tidal gravity as we observe and experience it could not exist. But since it evidently does, we know that time (as the concept is used in physics) is quite real, at least in the classical domain. On quantum scales on the other hand, the issue is more subtle and rather less straightforward - a case could potentially be made for time to not be fundamental on small enough scales. However, that would make it emergent, and still not an “illusion”; again, an important difference.

You are correct in the sense that there is no physical clock or ruler that could co-move along with a massless object - which is to say such objects do not have a valid rest frame. Nonetheless, the world line of a massless particle in spacetime does have a well-defined length, you just need to parametrise it using something other than proper time (which is zero by definition); so there is still a meaningful notion of massless particles travelling a certain distance in a certain amount of time, it just needs to be defined in a consistent way.

Or they may engage in masking - which is very energy consuming.

Precisely. +1

The Schwarzschild solution is static and stationary, and it requires some boundary conditions that one wouldn’t find in the real world for this type of event - so it is not a suitable model to describe the actual collapse process itself. It can only be used to approximately model either the end result or the original gravitating object, i.e. the original star or the resulting black hole, which are then assumed to be static and unchanging. Describing the collapse process itself is a very much more complex task, which can really only be done using numerical methods. The reason why the event horizon radius in Schwarzschild spacetime happens to agree with a suitable Newtonian calculation is that this type of spacetime is spherically symmetric and asymptotically flat. But this is not necessarily true in general - if you relax any of the boundary conditions of Schwarzschild (e.g. allow the central mass to carry angular momentum, or electric charge), the location of the event horizon will no longer agree with Newtonian theory. In fact, you will quickly find that there is in fact more than one horizon surface, giving quite a complex structure. The existence, nature, and location of horizon surfaces is intrinsically a relativistic phenomenon which has to do with the causal structure of the spacetime in question. It can, and (given certain conditions) inevitably must - but you have to use the correct model.

If you look carefully, you’ll notice that he hasn’t really addressed most comments made so far. He either ignores them, or just repeats earlier posts.

Yes, that’s not what I am trying to do - though I do think it is possible to a large extent to talk about gravitational physics without making reference to coordinates. But of course, you need a suitable choice of coordinates to extract numerical predictions from the model, I wouldn’t deny that at all. What I am attempting to emphasise is merely that the coordinate system doesn’t encode the actual physics on your spacetime, anymore than street names on a map encode the layout of a city; so while it might be necessary in practical terms, it isn’t fundamental as such. The physics are found in the metric and the connection. This is ultimately (in a physical sense) why one is free to change coordinate basis without affecting any physical prediction the model makes.

I’ve also pointed this out to him already on his “Norm of 4-Acceleration” thread, and gave the explicit expressions for the necessary transforms of x and ct, but he ignored it completely. In fact, so far he has ignored pretty much everything that has been pointed out to him - it doesn’t seem like there is any genuine openness to feedback. It feels more like a personal blog to me. Just for the casual reader’s reference - in the special case of motion starting from rest at the origin, with uniform acceleration, the integral becomes straightforward, and the expressions evaluate to: \[v( \tau ) =c \tanh\left(\frac{a\tau }{c}\right)\] \[v(t) =\frac{at}{\sqrt{1+\left(\frac{at}{c}\right)^{2}}}\] Which is, of course, hyperbolic motion, as we would expect in a Minkowski spacetime.

Yes, and it also doesn’t imply that Riemann is always identically zero for all manifolds. Which it evidently isn’t.

Well, in a pure mathematics context, the metric has a pretty abstract definition in terms of something called the “first fundamental form” - which essentially boils down to defining the notion of an “inner product” on your manifold. Physically, this means that a metric enables you to quantify lengths, areas, volumes, angles etc - so it makes it possible to relate physical laws to real-world measurement outcomes. More specifically in GR, it is used to quantify the separation between events in spacetime. Given a connection, this determines all the various curvature tensors etc. I must reiterate that it is not my intention to deny that some coordinate system needs to be chosen in order to actually perform the calculation - however, the point is that the physics aren’t in the coordinate system itself, they are in structure of the metric, i.e. in how the components of the metric tensor (in a given coordinate basis) are related to one another. This is an invariant property, unlike the value of the components themselves. I meant it in this sense. Indeed, that’s pretty much what I am attempting to point out, with the addition that in the context of GR, the metric is the more fundamental object. Indeed.

Ghideon has essentially said it already (+1) - gravity isn’t actually a force (except as an approximation in the Newtonian domain), so you can’t model it using a vector field approach, such as the one you have presented. It also exhibits different dynamics than electromagnetism, some superficial similarities notwithstanding. It is, in fact, possible to mathematically show that no vector field theory can accurately model gravity, since such models are intrinsically incapable of encoding the necessary degrees of freedom (for fundamental, albeit technical, reasons). You need at the very least a rank-2 tensor theory, such as GR for example.

Yes of course, that much should be obvious to all of us. It was not my intention to imply anything different. I meant my comment in the sense that the structural elements I had listed are what I think (!) is minimally required to construct the specific model of General Relativity, as opposed to other models of gravity. For example, change the Levi-Civita connection to a Weizenböck connection, and you get “Teleparallel Gravity”. So I meant it in that sense. To be honest, the list I gave was what spontaneously popped into my head when I read your previous post, I haven’t actually thought about it too deeply. You are right that diffeomorphism invariance is important, but rather than putting this as a separate requirement, I would amend (3) to say that every small enough patch is to be locally Minkowskian. Along with the automatic conservation of the Einstein tensor, this would imply both the correct metric signature, as well as diffeomorphism invariance, and the smoothly connects to SR as a limiting case. As for the coordinate system, I think this is already implied by the requirement to have a metric. The important point is that you can choose any suitable coordinate basis that fulfils basic analytic conditions of continuity and differentiability, so I think a choice of coordinates is secondary to the metric. One could easily formulate all of GR in abstract geometric language, without any reference to specific coordinates at all - as Misner/Thorne/Wheeler have done in their text “Gravitation” for example. Is it really? I would tend to think that it is only the metric that is essential, but not any specific coordinate system. What makes spacetime is its structure, not the labels we assign to each event. The conservation of the stress-energy tensor follows from Noether’s theorem, so it will locally hold whether gravity is described by GR or not. The question whether the automatic (!) conservation of the Einstein tensor is a necessary condition, is far more interesting. I think that it is, because by demanding this we essentially fix the general form of the local constraint on the metric (up to constants), i.e. the Einstein equations. If we don’t demand this, then we no longer have a clear mechanism to couple stress-energy to any specific notion of curvature, and the field equations could take other forms - for example, they could contain higher powers of Riemann and its derivatives (such as e.g. Lovelock gravity), or functionals of Ricci (such as f(R) gravity e.g.). So I think the requirement is necessary to uniquely recover GR, as opposed to other possible metric models. I think it is worthwhile also to note that this conservation requirement has an underlying deeper structure, being the topological principle of "the boundary of a boundary is zero". Again, Misner/Thorne/Wheeler have a very good presentation on this in "Gravitation". The point here is that this is not a completely arbitrary requirement. Spacetime is a mathematical model, it is not any kind of physical substance or object - so I don’t think “what is it physically made of” is necessarily a meaningful question. Nonetheless, if I was to attempt a classical answer, then I would say spacetime is a network of relationships between events, and those relationships are constrained in specific ways. In a deeper quantum sense, one could go with one of the newer ideas, for example that spacetime arises from entanglement relationships, or that it is a spinfoam network. Or perhaps one can look at it in terms of information. Upshot is, I think it is reasonable to (preliminarily) say that the ontology of spacetime is a mathematical model, whatever specific form it might take. This of course raises the question - is spacetime an actual feature of the physical world at all, or is it just the mind’s way to structure information and construct its PRM (phenomenological reality model - ref Thomas Metzinger)? Because that’s all our directly experienced “reality” is - a model constructed by the mind. How this maps to an external reality (if that is even a meaningful concept) is anyone’s guess. Indeed! Subjective phenomenology (=direct experience) being another example.

1. A topological manifold 2. The Levi-Civita connection 3. A metric with the correct signature 4. A local constraint on the metric which guarantees the automatic conservation of the Einstein tensor (=the Einstein field equations) This is pretty much the minimum structure required to get GR, as opposed to other models of gravity.

This doesn’t make any sense. The apparent motion of the sun through the sky in the course of a day is due to Earth’s rotation, not due to relative motion between Sun and Earth. So the Sun-Earth distance does not come into this at all. The issue of course is that Flat Earth rejects the notion of a rotating planet, so pointing this out will just result in hand-waving dismissal. My advice: don’t bother. I’ve been there too, and all it ever resulted in was unnecessary grief. That particular community rejects not only basic scientific observations (such as gravity e.g.), but even the very scientific method itself; there simply isn’t any common ground to base a meaningful debate on.

Let’s look very briefly at what the symmetries of Riemann actually constrain. We have two sets of symmetries - first, those that are present even in the absence of a metric: \[R{^\alpha}{_{\beta \gamma \delta}}=R{^\alpha}{_{\beta [\gamma \delta]}}\] \[R{^\alpha}{_{[\beta \gamma \delta]}=0}\] \[R{^\alpha}{_{\beta [\gamma \delta ||\mu]}}=0\] Second, we have metric-induced symmetries: \[R_{\alpha \beta \gamma \delta}=R_{[\alpha \beta]\gamma \delta}\] \[R_{\alpha \beta \gamma \delta}=R_{\gamma \delta \alpha \beta}\] \[R_{[\alpha \beta \gamma \delta]}=0\] What this means: in dimension n, every index pair can take on n(n-1)/2 independent values (due to their anti-symmetry), which initially leaves us with a symmetric matrix with \[\frac{1}{2}\left(\frac{n( n-1)}{2}\right)\left(\frac{n( n-1)}{2} +1\right) =\frac{n( n-1)\left( n^{2} -n+2\right)}{8}\] independent components. The Bianchi identities (last relation in non-metric symmetries above) constrain a further n!/(n-4)!/4! components. This finally leaves us with \[c_{n} =\frac{n^{2}\left( n^{2} -1\right)}{12}\] functionally independent components of the Riemann tensor. In n=4 dimensions, this evaluates to 20. So on a spacetime manifold with 4 dimensions, the symmetries of Riemann leave 20 tensor components unconstrained and functionally independent, meaning those components are not identically zero in the general case. Hopefully this clears things up, since it is trivially obvious that geodesic deviation does in fact exist in the real world (contrary to the OP’s claim), just like the theory says it does.

Indeed not. The cosmological model we are using (the Lambda-CDM model) is, at large scales, based on homogeneity and isotropy.