Markus Hanke

Gravitation in GR is geodesic deviation, and thus a geometric property of spacetime; all free-falling test particles experience gravity (they must follow geodesics in spacetime), regardless of whether they have mass or not, and regardless of their internal composition or size. Remember also that within the Standard Model, all fundamental particles are point-like, i.e. any mass distribution is simply a collection of point particles. Relative motion is not a source of gravity; the source term in the field equations is the stress-energy-momentum tensor, which, as being a tensor, is covariant under Lorentz transformations. If that were not so, the theory would not be internally self-consistent. It is important to reiterate that there are two physically distinct types of time dilation - there is kinematic time dilation due to relative motion (which also happens in flat Minkowski spacetime), and there is gravitational time dilation due to curvature of spacetime (which only happens when gravitational sources are present). These two effects can be present simultaneously, but they are nonetheless physically distinct effects.

I am honestly not sure if I follow your thought process correctly, since such a notion as “time dilation gradient” does not make much sense to me. But nonetheless, the aforementioned case of an orbit around a rotating mass should be an example. Another scenario that immediately comes to mind would be two parallel beams of light (or any other pp-wave spacetime, for that matter) - if you fire two parallel beams of light in the same direction, there will be no gravitational attraction between them, even though they carry energy. But if you fire the same two beams of light so that they are initially parallel, but travel in opposite directions (i.e. you let emitter and receiver trade places for one of the beams), then they will indeed experience a gravitational attraction. There will of course be time dilation between a clock inside the volume of dust, and some other reference clock outside of the dust cloud; but there is no time dilation between two clocks that are both located inside the dust cloud. I should have been more clear on this, as I was initially thinking of the cosmological case, where there is no “outside”. As I said, I don’t immediately recall where I saw that proof, it was a few years back when I came across it, and it was in a printed textbook. However, I can offer an outline (!) of my own attempt at proving this, for whatever it is worth. For this, allow me to go back to the basics, and consider what it actually means for a manifold (such as spacetime) to have curvature, and how to capture this mathematically. Imagine you choose some arbitrary point P on your manifold, and pick out an arbitrary tangent vector attached to that point. Now you parallel-transport that tangent vector around a small (i.e. infinitesimal) loop that starts and ends at your point P. The question is - will the initial vector before the parallel transport operation coincide with the final vector at the end of the procedure, regardless of the specific curve the loop describes, and what direction I travel on that loop? On a flat manifold, using standard calculus, the answer is obviously yes (I use single bars “|” to denote ordinary derivatives), since ordinary derivatives commute: \[A_{\mu |\nu \gamma } -A_{\mu |\gamma \nu } =0\] However, if we allow the manifold to not be flat, then the situation changes; following the standard prescription for this (refer to any textbook on differential geometry), we must now replace ordinary with covariant derivatives, which do not in general commute. The degree to which they fail to commute is (I use double bars “||” to denote covariant derivatives): \[A_{\mu ||\nu \gamma } -A_{\mu ||\gamma \nu } =R{^{\delta }}{_{\mu \nu \gamma }} A_{\delta }\] The object \(R_{\mu \nu \gamma \delta}\) is called the Riemann curvature tensor, and it uniquely specifies all aspects of the geometry of a given manifold. The question then becomes how you explicitly calculate the components of the Riemann tensor, i.e. what kind of object is it a function of? For this you need to only remember that GR uses the Levi-Civita connection, which is torsion free; this implies symmetry in the lower indices of the Christoffel symbols: \[\Gamma {^{\gamma }}{_{\mu \nu }} -\Gamma {^{\gamma }}{_{\nu \mu }} =0\] This being the case, you can then work out an explicit coordinate expression for the Riemann tensor from the above equations. I won’t typeset it here now since it is tedious to write in LaTeX notation (you can easily Google it, if you are interested) - I will simply point out that the Riemann tensor turns out to be a function of the connection coefficients and their derivatives only, which in turn are functions of the metric tensor and its derivatives only. So in other words, and that is the point of this whole exercise, given the fact that GR uses the Levi-Civita connection to describe parallel transport, a unique description of all relevant aspects of a manifold’s geometry under GR (i.e. the Riemann curvature tensor) arises from a rank-2 tensor, being the metric tensor. A simple accounting of the indices in the above expressions show that there is no mathematical possibility of any lower rank object (such as a scalar or vector) doing the same job. Which is what we wanted to show. The above is obviously only an outline - you could fill in the details and actual calculations yourself, using any standard textbook on differential geometry. I don’t know how rigorous the above really is, but that’s how I would approach such a proof - quantify the failure of derivatives to commute on curved manifolds; then, given a connection, check what kind of object the coordinate expression for the curvature tensor depends on. It seems pretty simple and logical to me. But if someone here who is actually an expert in the area can think of a better, more rigorous way, or can point out an error in the above reasoning, then I would definitely be interested in seeing it!

Well, you can consider a hollow sphere made from a thin shell of matter. The exterior of the shell looks like any other spherically symmetric body, so it is described by the usual Schwarzschild metric. The hollow interior of the shell however is a different story - no tidal gravity is detected therein, meaning a test particle placed anywhere into the interior remains at rest. At the same time though, if you place a clock into the interior, and somehow compare its tick rate against a reference clock far away on the outside, you will find that it is time dilated, even though no forces (which would cause it to move) are detected locally where the clock is. So this would be an example of time dilation, but no tidal forces. Another even simpler example would be a uniformly accelerated frame in an otherwise empty region of spacetime; again, an accelerated clock is dilated, but there is no tidal gravity. A real-world example of a case where you have tidal gravity in the spatial part of the metric, but no time dilation, would be a region of spacetime that is uniformly filled with dust, in a way that ensures homogeneity and isotropy. The FLRW metric - on which our current understanding of cosmology, the Lambda-CDM model, is based - is an example of this. In this metric the temporal part is constant, but the spatial part is not. I should also mention here that within metrics, each coordinate coefficient can depend on all coordinates, including time. So not only can things vary as you move in space, they can also vary with time, and with any possible combination of the two. So you can get quite complicated spacetimes that are neither static nor stationary, with highly non-intuitive geometries. And if that wasn’t enough, then it needs mentioning that the dynamics of GR are highly non-linear, meaning gravity self-interacts; hence (at least in principle) you can have topological constructs that are formed and held together purely by their own gravitational self-energy, in the complete absence of any “traditional” sources. I think you are beginning to see now that the dynamics of spacetime are very rich and varied - they can’t be captured by just assigning some scalar field. I actually seem to remember having once seen a formal proof that a rank-2 tensor is the lowest rank object required to capture all dynamics of GR, I just can’t remember where I have seen it. If I come across it, I will post it here.

Because - as I have attempted to explain - time dilation is a relationship between distant clocks, whereas a field assigns a particular object (a tensor of spinor of any rank) to each local event in spacetime. You cannot point to an event in spacetime and say “I am going to assign time dilation factor X to this event”, without any further qualification - this does not make any physical sense. The most fundamental entity in GR (and the solution to the Einstein field equations) is the metric tensor field - it assigns a metric tensor to each event in spacetime. To put it in the simplest possible terms, the metric tensor field allows you to quantify how each event in spacetime is related to all other events - both in spatial terms, and in terms of time. It does so by defining a mathematically precise relationship between neighbouring events, so that, by integrating along curves, you can calculate relationships between more distant events, e.g. the length of a world line connecting them. Time dilation in GR is a geometric property of world lines, in that it is the ratio between the lengths of world lines between the same events - the total time a clock accumulates between two given events is equivalent to the geometric length of the world line traced out by that clock. And how long that world line will be depends on the geometry of the spacetime it is in, and what kind of world line it is. Take for example a rotating spherical body, such as a planet. If you let a test clock orbit the planet once in its direction of rotation, starting and finishing at some point P, then that orbit will take a total time T1. If you now start at the same spot P, but orbit in the opposite direction (counter the planet’s direction of rotation, but along the same orbit, with all other initial and boundary conditions remaining equal), you will get some orbital time T2, which will be ever so slightly different. That’s because, even though you start at the same point P, and traverse the same spatial distance along the same orbit, the geometry of spacetime is such that the lengths of the two world lines will differ. The ratio between these two geometric lengths is one example of gravitational time dilation - the value of that ratio depends on where the point P is, the initial and boundary conditions of the clock kinematics, and the global geometry of the underlying spacetime. How would you capture all this by assigning a single value to point P, as you seem to want to do with your “time dilation field” idea? Again, on closer consideration, in order to capture all relevant degrees of freedom so that all aspects of gravity can be correctly modelled, independently of the precise circumstances, at least a rank-2 tensor field is necessary. That’s what GR does.

Time dilation is a relationship between distant clocks (not a property of them), wherein ‘distant’ just means that the clocks are separated in space and/or in time. Time dilation arises from the metric, which is a particular solution to the field equations for given initial and boundary conditions. No - again, because it is a relationship between clocks, not a fixed value that can be assigned to a given event.

GR uses the Levi-Civita connection, so there is no torsion. Also, if there were any vector fields involved, then those would be 4-vectors, not 3-vectors; and two separate vector fields still do not capture the necessary degrees of freedom. The field equations - like all physical quantities in GR - need to be covariant, so no, you can’t make any kind of explicit reference to an observer. One could also think of it in terms of gravitational radiation fields. These fields extend to infinity, and wave fronts in free space propagate at the speed of light; furthermore you have two distinct polarisation modes. In terms of field quanta, this automatically implies massless spin-2 bosons - which, mathematically speaking, can only “couple” to rank-2 tensors. So this is the lowest rank object that is needed to fully capture all relevant degrees of freedom of gravity.

Ok, so the gradient would give you a vector field - which is still insufficient to capture all the necessary degrees of freedom. As I said, at the very least you need a rank-2 tensor field to adequately describe gravity. Time dilation is a relationship between clocks, it’s not a covariant quantity, and it isn’t local either. So, such a thing as a “time dilation field” does not make much physical or mathematical sense. A solution to the Einstein equations is given by a metric - this is the primary and most fundamental mathematical object in GR. Once you have the metric, you can then calculate the relationship between given clocks in spacetime from this.

The concept of “gravitational potential” can only be meaningfully defined in some spacetimes with very specific symmetries; it, too, does not generalise.

It is not possible to capture all of gravity’s degrees of freedom with a scalar field theory (or even a vector field); you do require at the very least a rank-2 tensor field to do so. Even if you take just the next “baby step” up from Schwarzschild spacetime to Kerr spacetime, you will find that this concept no longer works. At least in principle you can still derive closed expressions for the relativistic optics in such a spacetime (though the maths are anything but trivial), but they no longer correspond to anything resembling a scalar time dilation field. I invite you to try it out, but be warned - some heavy maths ahead! And the whole thing most certainly does not generalise to arbitrary spacetimes.

This is valid only for spherically symmetric, non-rotating and uncharged gravitational sources in an otherwise empty universe, i.e. in spacetimes that are approximately Schwarzschild. It cannot be generalised to any other case, which is why it is not suitable as a general model of gravity. General Relativity on the other hand represents a general constraint on the metric, i.e. it constrains what form the geometry of spacetime can take, given appropriate initial and boundary conditions. It thus works as a model for gravity regardless of the precise nature of its sources, in any given purely classical scenario.

Don’t forget though that GR as a model does not stand in isolation - the large-scale physics of the universe need to remain compatible with the small-scale physics of the Standard Model. Unfortunately the QFD and QCD parts of the Standard Model Lagrangian are not scale invariant, so you cannot replace a universal expansion with a contracting observer, without breaking some crucial physics in the process.

Surely you meant to write \(g_{\alpha \beta}\), since \(G_{\alpha \beta}\) denotes the Einstein tensor, which is a different quantity.

Recasting the theory of electromagnetism into a geometrical form is straightforward if you use the differential forms formalism. It then becomes simply \[dF=0\] \[d\star F=4\pi \star J\] This has already been known for a long time. For a (very) detailed discussion on the similarities and differences between electromagnetism and gravity when it comes to their respective formalisms, I refer you to Misner/Thorne/Wheeler, Gravitation, chapter 15, most especially box 15.1. All of this has already been recognised and worked out in detail. Essentially, the form of both models shares a common underlying principle, being the topological principle that “the boundary of a boundary is zero”; but because the basic objects involved are different ones, you end up with two models that also have a lot of differences. In spite of any similarities, electromagnetism does not work the same way as gravity does. I would really urge you to consult the above reference, since it seems to me that what you are trying to do is something that has already been done long ago.

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