Markus Hanke 487 Posted December 7, 2019 Share Posted December 7, 2019 (edited) 5 hours ago, studiot said: Wouldn't it just be different? It wouldn’t be spacetime, because there would be no concept of distance in space or separation in time. 5 hours ago, studiot said: Extension (distance, interval) is now fundamental; and the location of an object is a computational result 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. 4 hours ago, studiot said: A very emphatic no I'm afraid. Dropping the coordinate idea of contours or isolines (t = a constant) is the most important idea both Marcus and Eddington stress. The idea of t = a constant is dangerously close to leading towards an absolute coordinate system - an anathema to relativity. 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. Edited December 7, 2019 by Markus Hanke Link to post Share on other sites

Mordred 1372 Posted December 7, 2019 Author Share Posted December 7, 2019 3 hours ago, geordief said: These invariant intervals ,can it be usefully said that they are not subject to any uncertainty in their measurements? You would deal with the position and momentum uncertainty with quantum particles however relativity itself is a classical theory which it's mathematics doesn't incorporate probabilities or harmonic oscillators for the uncertainty principle. Those get incorporated when you deal with theories such as QFT. However freefall paths via principle of least action (Langrangian) does involve uncertainty in the chosen path that the particle will take at each infinisimal. (GR itself doesn't get too much into the Langrangian) so when studying GR I wouldn't worry about uncertainty in freefall paths. That's would be far too distracting until you get really comfortable with GR Link to post Share on other sites

studiot 2146 Posted December 8, 2019 Share Posted December 8, 2019 23 hours ago, Markus Hanke said: 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. So have the boffins got this to work yet, with or without "auxiliary fields" ? Quote Quote Wikipedia ADM energy and mass See also: Mass in general relativity ADM energy is a special way to define the energy in general relativity, which is only applicable to some special geometries of spacetime that asymptotically approach a well-defined metric tensor at infinity – for example a spacetime that asymptotically approaches Minkowski space. The ADM energy in these cases is defined as a function of the deviation of the metric tensor from its prescribed asymptotic form. In other words, the ADM energy is computed as the strength of the gravitational field at infinity. If the required asymptotic form is time-independent (such as the Minkowski space itself), then it respects the time-translational symmetry. Noether's theorem then implies that the ADM energy is conserved. According to general relativity, the conservation law for the total energy does not hold in more general, time-dependent backgrounds – for example, it is completely violated in physical cosmology. Cosmic inflation in particular is able to produce energy (and mass) from "nothing" because the vacuum energy density is roughly constant, but the volume of the Universe grows exponentially. Application to modified gravity By using the ADM decomposition and introducing extra auxiliary fields, in 2009 Deruelle et al. found a method to find the Gibbons–Hawking–York boundary term for modified gravity theories "whose Lagrangian is an arbitrary function of the Riemann tensor".^{[6]} Link to post Share on other sites

Markus Hanke 487 Posted December 8, 2019 Share Posted December 8, 2019 2 minutes ago, studiot said: So have the boffins got this to work yet, with or without "auxiliary fields" ? 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). Link to post Share on other sites

studiot 2146 Posted December 8, 2019 Share Posted December 8, 2019 (edited) 4 minutes ago, Markus Hanke said: yes, it works fine, it’s just a different way to formulate the same model (GR). Despite the middle paragraph of the Wiki quote? here is the full reference. https://en.wikipedia.org/wiki/ADM_formalism I also don't see how using a t axis like that is compatible with the "Principle of Relativity" Edited December 8, 2019 by studiot Link to post Share on other sites

Markus Hanke 487 Posted December 8, 2019 Share Posted December 8, 2019 (edited) 28 minutes ago, studiot said: Despite the middle paragraph of the Wiki quote? 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? 28 minutes ago, studiot said: I also don't see how using a t axis like that is compatible with the "Principle of Relativity" 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. Edited December 8, 2019 by Markus Hanke Link to post Share on other sites

studiot 2146 Posted December 8, 2019 Share Posted December 8, 2019 (edited) On 12/7/2019 at 12:00 PM, geordief said: Looked at differently*, is it possible (in the model) to abstract space from spacetime by using regions of spacetime where t equals a constant.? On 12/7/2019 at 4:32 PM, Markus Hanke said: Actually, it is possible to describe spacetime as an ordered set (called a foliation) of spacelike hyperslices, where t=const for each slice. How is this possible without an underlying coordinate system ? On 12/7/2019 at 4:32 PM, Markus Hanke said: It wouldn’t be spacetime, because there would be no concept of distance in space or separation in time. I can't agree with this since the sticks (intervals) have a clearly defined measure. And clearly there exists a stick between each pair of events in the set. Even if the set includes every number in [math]\Re \otimes \Re \otimes \Re \otimes \Re [/math] Edited December 8, 2019 by studiot Link to post Share on other sites

Markus Hanke 487 Posted December 8, 2019 Share Posted December 8, 2019 (edited) 46 minutes ago, studiot said: How is this possible without an underlying coordinate system ? 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. 46 minutes ago, studiot said: I can't agree with this since the sticks (intervals) have a clearly defined measure. 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. 47 minutes ago, studiot said: And clearly there exists a stick between each pair of events in the set. Only if we have a manifold endowed with a connection and a metric, otherwise not. So we need that extra structure. Edited December 8, 2019 by Markus Hanke Link to post Share on other sites

studiot 2146 Posted December 10, 2019 Share Posted December 10, 2019 On 12/8/2019 at 5:44 PM, Markus Hanke said: 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. Having a metric is not an essential requirement for topological spaces. If a topological space has a metric it is a metric topological space. This is important because there is no requirement to measure the 'length' of the sticks in a topological network of connected sticks. The connectivity is all important in determining precedence or causality. So I maintain it would just be different, although topologically equivalent. I do agree that if you restrict the use of 'spacetime' to Minkowski (who coined the word after all) then it would not necessarily be spacetime. Note also that in the first millenium and a half before coordinate systems were invented Geometry functioned perfectly well. In fact the introduction of coordinate systems, principally by Descartes, introduced extra information into Geometry which was not present before. This extra information is that everything now has an orientation. Before an equilateral triangle was the same whichever way up it presented. Whereas the same triangle standing on a vertex or a base are considered to be different different. The issue then becomes is this redundant or required information ? There is a move in modern Geometry to return to the pre Descartes era. Link to post Share on other sites

Markus Hanke 487 Posted December 11, 2019 Share Posted December 11, 2019 18 hours ago, studiot said: Having a metric is not an essential requirement for topological spaces. 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. 18 hours ago, studiot said: The connectivity is all important in determining precedence or causality. 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. 18 hours ago, studiot said: I do agree that if you restrict the use of 'spacetime' to Minkowski (who coined the word after all) then it would not necessarily be spacetime. 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. 18 hours ago, studiot said: The issue then becomes is this redundant or required information ? 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. Link to post Share on other sites

dad 68 Posted February 19, 2020 Share Posted February 19, 2020 On 5/30/2015 at 7:11 PM, Mordred said: This question is amongst one of the most commonly asked questions in relativity. I guess the answer is science doesn't know. Why guess? -2 Link to post Share on other sites

beecee 542 Posted February 19, 2020 Share Posted February 19, 2020 1 hour ago, dad said: I guess the answer is science doesn't know. Why guess? Theories are not guesses. They are best estimates at any specific time, that may or may not change, as technology advances and more data becomes available. Those same theories grow in certainty as they continue to match the observational and experimental data: The theory of evolution of life is actually a fact...other theories such as the BB, SR/GR are overwhelmingly supported. But didn't I explain this to you elsewhere earlier on? Glad to be of further assistance anyway. Link to post Share on other sites

Phi for All 5884 Posted November 28, 2020 Share Posted November 28, 2020 ! Moderator Note Alternative explanation and discussion split to here. Link to post Share on other sites

Col Not Colin 9 Posted January 26 Share Posted January 26 New User here. Hi everyone. Please advise me where to go and what is appropriate, my apologies in advance for getting it wrong so far. 1. Thanks for pinning this post to the top of the board, it seems like a good one for newbies to start with. 2. It would help if the original creator(s) posted an update, so that we could see what the article looks like after 4 years of comments and editing. I've only had a scan through all the comments and that took a whole day. 3. Are you sure people often ask "What is space made of"? Have you got any surveys or other evidence for that? In my limited experience, most are happy to assume it's an empty vaccum. However, I do agree that people are quicky and frequently mis-lead into thinking that space is a stretchy fabric. I also agree with many of the later comments about quantum foam and vaccum energy that you could try to incorporate into your definition of space. If you are determined to incorporate QFT then I suppose spacetime does have to have fields in it. The problem is being certain that you have covered everything in your attempt to say what space is made of or even what mathematical objects must be supported and exist in space. There is almost no way of future-proofing the idea. Early General Relativity only really considered simple particle theories. It's unlikely anyone knew that anything like QFT, dark energy or vaccum energy could be added later. It's amazing that GR still seems to hold so well given that we have incorporated so much more into it than it was originally designed to model and we would be extremely optimistic to assume that nothing more will ever be added. As a consequence, I suspect the original creator would be better off not trying to tell people what space is actually made of - but instead focuses on what is the least amount of structure required for GR. Therefore, his/her definitions of spacetime as a (pseudo) metric space seem the better way to go. It is an ambitious target to try and describe all the (intangible) objects, like fields, that may be inferred or required to exist in space. 1 Link to post Share on other sites

Markus Hanke 487 Posted January 27 Share Posted January 27 10 hours ago, Col Not Colin said: but instead focuses on what is the least amount of structure required for GR 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. 1 Link to post Share on other sites

studiot 2146 Posted January 27 Share Posted January 27 (edited) 18 hours ago, Col Not Colin said: Are you sure people often ask "What is space made of"? ..... As a consequence, I suspect the original creator would be better off not trying to tell people what space is actually made of - but instead focuses on what is the least amount of structure required for GR. Therefore, his/her definitions of spacetime as a (pseudo) metric space seem the better way to go. It is an ambitious target to try and describe all the (intangible) objects, like fields, that may be inferred or required to exist in space. Hello Col and welcome. +1 for a good start, I look forward to further worthwhile contributions. I've been a member since 2012 and in that time I have seen many threads started discussing the question, "What is space ?" So it is indeed apressing question. So pressing that in fact we now need to separate what is meant by a physical space and what is meant by a mathematical space. Defining a mathematical space is easy. You need a set containing at least three (perhaps four) sets of objects. 1) A set of mathematical objects you wish to work with. 2) A set of coefficients you wish to apply 3) A set of axiomatic relations between these objects 4) Perhaps if you want to be complete then a set of results (theorems lemmas etc) you can deduce from these. 8 hours ago, Markus Hanke said: 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. Hi Markus, I think you have this the wrong way round. Mathenmatical structures are models of physical reality, rather than physical reality being a model of mathematics. Edited January 27 by studiot 1 Link to post Share on other sites

Col Not Colin 9 Posted January 27 Share Posted January 27 Hi Markus. Thanks for your comments. You may be right in your collection of requirements. However, there is some over-lap in the properties that they explain and will require and it is obviously possible to identify alternative criteria that are sufficient. I hope you'll forgive me if I take a moment to determine if the total collection of requirments you presented really is sufficient structure for GR. There were some earlier comments about requring a co-ordinate system and I would be a bit relunctant to let them go. Text-book definitions of the structure required in GR do seem to utilise charts, an atlas, diffeomorphisms mapping the manifold to/from R^n etc.. Such a definition also has the advantage of providing your requirement no.1 automatically. There is a topology induced on the manifold from R^n (specifically R^n with the usual metric on R^n). There doesn't seem to be an obvious way to extract a co-ordinate system from the set of requirments you have listed. I would say a co-ordinate system is essential for what is defined as Spacetime in GR, otherwise what you have is a perfectly interesting object (a topological space with some extra structure) that may exhibit properties that are in some sense similar to spacetime and GR but it's not what we can define as Spacetime. Hi Studiot. Thanks and I'll get to your coments later. I've got stuff to do for an elderly relative first. Please advise me if non-scientific personal stuff should be kept off the forums. I'm also not too interested in scoring points but I'll look into how I can award points to others later if that's important to everyone else. Link to post Share on other sites

beecee 542 Posted January 27 Share Posted January 27 As a lay person with an interest in science and the scientific methodology, my perhaps simplistic view is as follows....Space is what exists between you and me, or between the planets and their stars. Time is what stops everything from happening together and the means by which the intervals between sequential events are measured. Spacetime is the four dimensional framework against which we locate and calculate events. The concept of spacetime follows from fact that the speed of light, "c," is constant and does not vary with the motion of the emitter or the observer. It is essentially a description of reality common for all observers, while at the same time, the measured Intervals of space and time when considered separately will vary between observers and different frames of references. Gravity of course as GR tells us, is the geometry of spacetime in the presence of mass/energy. All are essentially real: Space is expanding and the source of what we know as Dark Energy. Time is essentially interchangeable with space and is a variable quantity. Spacetime can be bent, warped, curved, lensed in the presence of mass. Any errors, alterations or corrections? Please be gentle with me. 😜 1 Link to post Share on other sites

Col Not Colin 9 Posted January 28 Share Posted January 28 4 hours ago, studiot said: Defining a mathematical space is easy. I agree. This is the main reason why I suggested the OP focuses on this. It is far more difficult to explain what space is physically made of and the best answer may be "don't know, don't care, it's not required knowledge for GR". Space and Spacetime must have this Mathematical structure (and present a list of requirements similar to Markus) but that is all. I've also had a bit more time to think about what the OP (I believe that stands for Original Poster) was trying to do and where the collection of comments have gone since then. I've had enough time that I might even retract my own earlier statement There isn't much need to focus on what mathematical structure spacetime should have. That's already clearly explained and set out in any text-book. What Markus has done is slightly more interesting: We may be able to shuffle the requirments around and demand some types of mathematical structure instead of others and there should be a minimal set of requirements (although, I'm sorry to say that I'm not convinced Markus' list is that minimal set because I cannot see how we are going to get a co-ordinate system out of that list of structures and properties). However, defining spacetime by the minimal set of mathematical structure that is defined upon it and/or the minimal set of mathematical properties it has, is probably only of interest to people who favour Mathematics. 6 hours ago, studiot said: in fact we now need to separate what is meant by a physical space and what is meant by a mathematical space. Studiot seems to be representing the viewpoint of a more general scientist and not a pure mathematician. Defining spacetime as a Mathematical structure does not explain what it is, only how it behaves. He also goes further and states that Physical reality is the key and Mathematical representation is not. 6 hours ago, studiot said: .....I think you have this the wrong way round. Mathenmatical structures are models of physical reality, rather than physical reality being a model of mathematics. Beecee seems to have a similar interest in defining what space is based on physical reality and if I've understood the essence of his comments correctly, there is another useful idea contained within: Start from Special Relativity and add just enough to model gravity as curvature. Which raises another minor challenge for Markus: Do we really need item 4 in your list - the conservation of the stress-energy tensor? Are you (Markus) talking about the Binachi identities, the field equations with zero curvature (all zero Rieman Tensors), or something else? If our goal was JUST to model gravity as curvature then do we need anymore than the geodesic equation and an assumption that particles travel along geodesics (I suppose we could summarise both of these by saying that we require a least action principle)? Perhaps having the Einstein Field Equations is not a required goal. Your other requirements (such as 1 and 3) imply that the metric has already been given, or at-least is assumed to exist, and so we don't necessarily need the Einstein Field Equations to determine the metric. Don't worry, Markus, I saw your last line about "GR as opposed to other models of gravity", I appreciate that what I've just described is not a full model of GR as we know it. The main point is, until the OP or the audience make it clear what it is and exactly how much they need explained, we can't be sure what the minimal amount of structure required will be. There was a person called dad who posted in February 2020 and said something that actually seems sensible enough to me: On 2/19/2020 at 7:11 AM, dad said: I guess the answer is science doesn't know. Why guess? (I don't seem to be able quote that one too well... the above was an answer to Mordred's original question "What is space made of"). I don't know the full context of dad's post but it was hated and got a -3 score. The only thing I would have added to it was ...because it doesn't matter, it's not required knowledge for GR. Obvioulsy I think we can and should discuss the issues otherwise I wouldn't be here but dad's post doesn't seem that silly to me. I'm going to put a +1 score on that. Anyway, is the OP still producing an article describing what space is made of? Is it time to focus on what space is actually physically made of instead of worrying about the mathematical structure it must have? What are we doing here and what is the purpose of this "thread" (that's what they used to call a collection of forum posts in the old days)? Returning to Studiot's comments. What is it that most people end up discussing in all those other threads entitled "what is space?" If the OP no longer has any direct goal and this thread stays open, what kind of definition or explanation is it that people want? Link to post Share on other sites

MigL 1493 Posted January 28 Share Posted January 28 Interesting but somewhat confusing. You want a differentiation between the mathematical description, and actual, physical space. You have stated we have the mathematical description, but haven't yet tackled what we mean by actual, physical space. IOW, we have the model, but what is the actual reality ? ( and Dad says we don't know, which you agree is valid ) I would assume, as a scientist, you know how science works. We build models ( mathematical models, not toys ) and test them against reality to find their limited areas of applicability. The only model that is 100% applicable is the reality itself. I have to ask, how else would you model actual, physical space, if not mathematically ?? 1 Link to post Share on other sites

Col Not Colin 9 Posted January 28 Share Posted January 28 (edited) Hi MigL. I'm an old person, we used assembly language on a computer not modern internet acronyms. Does IOW = in other words? Anyway, I don't especially want anyhting. I'm not sure what the thread wants or what the OP wants. I consider myself a scientist, although my background is mainly in Mathematics. I almost agree with what you said about how science works. I, personally, wouldn't model physical space any other way. However I do have an interest in what space is actually physically made of, in addition to what physical and mathematical properties it must have - but that's not the main point, what does this thread or the OP want? 2 hours ago, MigL said: I have to ask, how else would you model actual, physical space, if not mathematically ?? For the sake of starting what may be a new discussion: There are other ways you could model the physical universe. See, for example, the work of Stephen Wolfram who uses ideas from computation. I do NOT mean programming a computer to find numerical solutions to some Mathematical model but instead modeling the universe as if it is governed by some simple, small algorithms from which he does seem to identify some emergent structure which can be recognised as General Relativity. I'm NOT an expert on any of this and I may have mis-represented his work but the essential idea is that it is not like building a mathematical model as we would understand it. Late Editing: I'm an idiot, sorry. MigL was probably talking to the OP. Edited January 28 by Col Not Colin Link to post Share on other sites

Markus Hanke 487 Posted January 28 Share Posted January 28 (edited) 16 hours ago, studiot said: Mathenmatical structures are models of physical reality, rather than physical reality being a model of mathematics. 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. 15 hours ago, Col Not Colin said: There doesn't seem to be an obvious way to extract a co-ordinate system from the set of requirments you have listed. 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. 15 hours ago, Col Not Colin said: I would say a co-ordinate system is essential for what is defined as Spacetime in GR 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. 7 hours ago, Col Not Colin said: Do we really need item 4 in your list - the conservation of the stress-energy tensor? 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. 7 hours ago, Col Not Colin said: Is it time to focus on what space is actually physically made of 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. 4 hours ago, Col Not Colin said: There are other ways you could model the physical universe. Indeed! Subjective phenomenology (=direct experience) being another example. Edited January 28 by Markus Hanke Link to post Share on other sites

joigus 370 Posted January 28 Share Posted January 28 8 hours ago, Col Not Colin said: Hi MigL. I'm an old person, we used assembly language on a computer not modern internet acronyms. Does IOW = in other words? Hope this helps. 2 Link to post Share on other sites

MigL 1493 Posted January 28 Share Posted January 28 I'm an old person too, Col. yes, IOW = in other words 3 hours ago, Markus Hanke said: Is it really? I would tend to think that it is only the metric that is essential, but not any specific coordinate system. I believ I was the one who originally mentioned the need for a co-ordinate system, back on page 11 of this discussion. Maybe my idea of a metric is not the same as yours Markus, and if not please elaborate so that we may all understand better. Is the metric not a measure of the deviation of the co-ordinate space ? It surely isn't a measure of actual deviation in 'real' space. (IOW, space-time has no actual 'fabric' that bends ) 1 Link to post Share on other sites

dimreepr 1077 Posted January 28 Share Posted January 28 4 hours ago, MigL said: I'm an old person too, Col. yes, IOW = in other words I believ I was the one who originally mentioned the need for a co-ordinate system, back on page 11 of this discussion. Maybe my idea of a metric is not the same as yours Markus, and if not please elaborate so that we may all understand better. Is the metric not a measure of the deviation of the co-ordinate space ? It surely isn't a measure of actual deviation in 'real' space. (IOW, space-time has no actual 'fabric' that bends ) IOW, down-hill is just a direction... 1 Link to post Share on other sites

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