Markus Hanke

Indeed not, but it is an essential requirement in order for said topological space to be considered a spacetime manifold, i.e. a model we can extract quantifiable physical predictions from. Yes, absolutely. In GR, the connectivity (i.e. relations between tangent spaces at different points) is given by the Levi-Civita connection, and the metric provides a way to define measurements. I am unsure whether we are talking about the same thing here now. In order for a given manifold to be a spacetime manifold in the sense of GR, it has to be endowed with both a connection and a metric, or else we are no longer doing GR. Of course, purely mathematically speaking, you can have manifolds without a metric, and these can be studied (ref differential topology), but then you can’t assign a consistent notion of length to curves on this manifold. This makes them rather useless, in terms of extracting physical predictions from them, other than general statements of topology. Well, I guess that depends on what it is you are trying to model with these manifolds. Within GR, we want to be able to study relationships between events, and quantify those in a consistent manner. For that purpose, you do need both a connection and a metric. For other purposes, a connection alone might be sufficient.

You do have to impose a coordinate system to define a foliation (which mathematically is just a set of functions of the metric), but you are free to choose whichever coordinate system works best for the problem at hand. There is no physically preferred one. So different observers are free to choose different foliations for the same scenario, but these will be related via diffeomorphisms, so they describe the same spacetime. This is the exact same situation as standard GR, just written differently. I agree that we need ‘sticks to connect the events’ - that’s really what I was trying to say all along, just in different words. You need to endow your manifold with a connection and a metric, before you can define a (quantifiable) notion of separation between events. Without that extra structure (connection & metric), you have a set of events, but no way to meaningfully define separations in time and space, nor indeed any kind of causal structure. So it wouldn’t be spacetime as we experience it, because it would lack any structure, geometry, or topology. In GR, this is done by endowing the underlying manifold with the Levi-Civita connection, as well as the metric as dynamic variable constrained by the Einstein equations. That is why, when we perform actual calculations in GR to do with separations in time and/or space, these are always based on the metric. All I am really trying to say here is that a collection of events alone does not constitute ‘spacetime’ - you need a connection and a metric structure as well (which would correspond to the ‘sticks’ you mentioned) to define meaningful relationships between these events. You need sticks to connect events, in your words. Without this, I’m pretty sure you wouldn’t even have a manifold in the mathematical sense, because there is no locally defined affine structure to the set (open to correction on this point, though). It seems to me that we are actually in agreement on this point, we are just explaining it in different ways. Only if we have a manifold endowed with a connection and a metric, otherwise not. So we need that extra structure.

ADM energy is just one of many different concepts of energy you find in GR; it applies only to some very specific types of spacetime, but can be quite useful in those cases, since it is relatively straightforward to calculate. What issue specifically do you see with this? Foliating a region of spacetime into space-like hyperslices is not the same as postulating an “absolute time” axis, because there are infinitely many possible foliations. In practical terms, you can label the slices in whichever way is suitable for the given problem at hand, there is no physically preferred foliation scheme, so there is no issue with the principle of relativity. The overall model retains full diffeomorphism invariance. Just to make this clear, the ADM formalism is just a different mathematical formalism of the same theory of GR - it has all the same symmetries, makes the same predictions, and has the same physical content. It’s simply a straightforward application of the Hamiltonian framework (a commonly used a very useful tool) to GR; so you just use a different set of dynamic variables to describe the exact same thing. It is particularly useful, and routinely used, in numerical GR. P.S. If you are interested in the precise details of how this works, then Misner/Thorne/Wheeler “Gravitation” devotes an entire chapter to this formalism. Well worth a read.

I am not familiar with that particular work, so I can’t comment on it. But as for the ADM formalism itself - yes, it works fine, it’s just a different way to formulate the same model (GR).

It wouldn’t be spacetime, because there would be no concept of distance in space or separation in time. I haven’t read Eddington, but I agree with this quote. This is what I meant when I said that a collection of events without any additional structure could not manifest as spacetime in the way we experience it. So in that sense, relationships between events are more fundamental (in terms of physics) than the events themselves. Actually, it is possible to describe spacetime as an ordered set (called a foliation) of spacelike hyperslices, where t=const for each slice. The result is somewhat like the pages in a book - each page represents a snapshot of 3D space, and is labelled by a number, which plays the role of time. There is a well defined sequence of page numbers, corresponding to the arrow of time. Or you could think of it as the frames in a movie. This is called the ADM formalism, and allows you to write GR in terms of Hamiltonian dynamics. Both the (non-constant) separation between hyperslices, as well as the spatial geometry of the slices themselves, make up the curvature of spacetime. The ADM formalism is very useful in numerical GR, as well as in the mathematics of some models of quantum gravity.

It seems evident that if you had just a collection of events, without any causal relationships between them, then there would be no concept of 'spacetime' at all. So in that sense I agree, it is that interwoven network of relationships that turns a collection of events into a physically useful spacetime manifold. In practical and classical terms, you can put neighbouring events infinitesimally close together, and mathematically represent their relationships by endowing the manifold with a suitable connection and metric - which is pretty much what GR as a model does. I emphasise again that this is a useful mathematical model, a map of the terrain so to speak, not a physical something to be found 'out there'. It's really important to understand this.

Greetings everyone, I am back 😎 So what is space made of? I think we need to first upgrade the question a bit and ask: what is spacetime made of? The answer to this is that it is a collection of events, to be understood in the sense in which the term is used in physics. To be even more exact, it is made of causal networks of events, i.e. events plus information about how those events are causally related. In terms of GR this is described as a manifold with its intrinsic geometry. Space on its own would then be just the spatial part of that network. So essentially, spacetime is a way to structure and organise information. Looking at it this way opens up some interesting questions, not all of which fall under the remit of physics: exactly what kind of information underlies this concept? Can this same set of information be structured/modelled in other ways as well? Is this structure intrinsic to the information, or is it something we impose more or less arbitrarily? Etc. P.S. It is important to remember that spacetime isn’t a physical “thing”, rather, it’s a mathematical model that captures certain aspects of the universe. It’s like a map we draw of a given territory.

No such theory is possible, because EM dynamics are linear, whereas gravitation is not. They are fundamentally of a different nature. It depends what you mean by “chaos”. GR Gravity is completely deterministic, since it is a purely classical theory, but it is not always indefinitely predictable. Since gravity is highly non-linear, under certain circumstances you get chaotic systems - here “chaotic” is used in the sense that the evolution of such systems is highly sensitive to initial conditions. Even tiny perturbations of the initial conditions can have large consequences in long-term evolution of the system. This is a well known phenomenon, which is found in many other areas of physics as well. I don’t understand what you mean by this...? No instantaneous actions at a distance can occur in nature. You can only have non-local correlations, which is a different thing, because that does not allow for the exchange of information. Electromagnetism is completely local, there are no non-local interactions.

He was unsuccessful because gravity is not an electromagnetic phenomenon. His approach was basically upside down. This is not entirely true. It is in fact possible to combine GR and EM into a single, overarching model, called Kaluza-Klein gravity. The problem with this is that it can’t be done in 4 dimensions, and also that it requires extra fields for which there is no evidence in the real world.

In post #7, under the metric tensor paragraph, it should read [math]g_{\mu \nu}[/math], and not [math]G_{\mu \nu}[/math], to avoid confusion with the Einstein tensor. Just a small thing though