Markus Hanke

We only really have good models up to and including the electroweak epoch. Anything before that is subject to uncertainty and speculation, even the GUT epoch. We have several candidate models for a GUT, but no consensus about which one - if any - correctly describes our universe. So this is still a subject of ongoing research.

This is true, but I think geordief asked his question in the context of GR, which is explicitly a metric theory of gravitation. My response was intended to remain in that same context.

“Global” in this context stands in contrast to “local”, and just means an extended region of spacetime (but not necessarily all of spacetime), which of course does have geometry - which is to say it is everywhere endowed with a metric. You do in fact need to “go global” to some degree in order to see GR effects, because every small enough local region will just look like the Minkowski spacetime of SR. Yes, you can apply GR to the universe as a whole. The FLRW solution is an example for this. I am not sure we are really talking about the same thing now, but cosmological solutions are more than just thought experiments. Standard cosmology, i.e. the Lambda-CDM model, is an example of a global/cosmological solution. Essentially yes, that is what is meant. This can be an extended region of spacetime, or even the universe as a whole.

We habitually take pain and suffering to go hand-in-hand, but they are not the same thing at all. Pain is a physical (or mental) sensation, whereas ‘suffering’ is how we react to that sensation - we experience it as something unpleasant, so we resist the experience and try to push it away. The stronger the pain sensation, the more we resist and thus suffer. As @nevimhas correctly pointed out, while we cannot stop ourselves from feeling pain, it is possible to train the mind out of its habitual habit patterns of resisting that unpleasant experience, and thus separate the suffering from the pain. There are specific training methods for this (i.e. meditation techniques), which lead to the ability to meet any experience - whether pleasant, unpleasant, or neutral - with complete equanimity (i.e. acceptance and non-resistance, not indifference). There are also other techniques that aim to block out the pain; those are deep absorption states, where, if you can get into them, the pain is no longer consciously felt. It has to be noted though that this type of training is not a quick process - for most people this takes a long time (often a lifetime) of consistent work to master.

Just to clarify something here: spacetime is the gravitational field. There is no gravitational field separate from spacetime - gravity is a geometric property of spacetime. So you have spacetime, which makes up the entirety of the universe, and its local and global geometry, which manifests as “gravity” in the way we experience it.

Before you even try to look into the particulars of QCD, I think it would be wise to learn about quantum field theory in general first (QCD being a quantum field theory). If you understand the basic principles of the underlying framework, things will become easier for you. Fair warning though - QFT is a notoriously difficult subject, and QCD a pretty complicated theory. But you’ll be able to pick up the gist of it quite easily.

Very well put.

No, the photons mediate the electromagnetic interaction between quarks, because quarks carry electric charge as well in addition to colour charge.

No. You cannot shoehorn QCD into a single particle. Due to the underlying symmetries required to make the (quite complicated) dynamics consistent, you need six separate quarks (along with their antiparticles) plus the gluon and the photon to make it all work. All of these are separate fields in the Lagrangian. The article you referenced has nothing to do with reducing QCD down to a single quantum field. Frankly (without intending any disrespect towards you), I would be very surprised if you truly understood this very technical paper.

Once someone has decided that the Earth is flat (or that any of the other ones is true), it is extremely difficult - and in many cases impossible - to get them to see reason. I have been on science forums for a long time now, and never once that I can remember have I seen an adherent to fringe ideas say: “You know what? You guys are right. I got this all wrong...”. It just doesn’t happen, except in extremely rare cases. Generally speaking such debates will quickly deteriorate into ad-hominems and mutual ridicule - leaving both parties with even more deeply entrenched ideas. This is counterproductive. Also, there is some form of social compartmentalisation happening - once someone has chosen to believe something, they will tend to surround themselves with people who share these beliefs, and shut themselves off from dissenting opinions. You get closely-knit, almost cult-like communities. So in my opinion the only way to combat this effectively is to stop as many people as possible from taking on such beliefs in the first place - stop the virus from spreading. An intellectual vaccination campaign, if you so will. Education plays a big part here, but is certainly not the only factor - I personally know a highly intelligent person, who has earned a university degree in philosophy with academic distinctions, yet is a passionate believer in pretty much all and any conspiracy theory out there. He just accepts them unquestioningly, even though I am sure on an intellectual level he must know they are nonsense. It seems people take on such beliefs in order to fill some void in their lives, to make themselves feel secure and in control in the face of an uncertain and menacing world. I don’t know, I’m not a psychologist, but I think that the problem is a socio-psychological one, not an intellectual one. From a scientific point of view, most of these ideas do not actually need debating, as they are self-evidently in violation of basic principles. Debating them might even give the impression that they have some value, and need to be refuted by science - so this can perchance do more harm than good. The other problem I see is that many people seem to lack the ability to distinguish valid sources and information from false ones. There is a pervasive attitude of “it came up on my Facebook timeline, so it must be true”. This is a major contributing factor. People must learn to check facts and distinguish valid sources from questionable ones - which I believe should feature in our school curricula in some form, as it is a skill that can at least to some degree be learned.

I think it would be helpful to take a step back and survey this issue from a wider perspective. First of all, a UFO is just precisely that - the designation for a flying object that for some reason or another has not yet been identified. Not more and not less. That does not mean that such an identification won’t still happen, and it does not mean that there is anything untoward, extraordinary, or supernatural going on. As such, UFOs are definitely real, since the capabilities of our visual and radar systems are necessarily limited, and subject to a lot of variables. Secondly, if the task is about finding evidence that UFOs are of extraterrestrial origin (which is often implied, but seldom explicitly stated), then the only permissible evidence will be to physically obtain an actual specimen, and study this according to the scientific method, i.e. in a transparent, peer-reviewed, repeatable and open manner. No visual sighting, radar contact, or even encounter - regardless of how convincing those may at first seem - will ever have any kind of scientific value in this regard, because there are too many variables present in such occurrences. Strangely enough, while there have been a mind boggling number of UFO sightings over the past few decades in particular, not even a single shred of publicly available physical evidence - in the form of a physical specimen that cannot be attributed to terrestrial sources - exists. Of course, there are many claims of debris having been found etc, but when you try and actually track down those objects, they all have mysteriously disappeared, or were allegedly removed and hidden by authorities, or the reports turn out to be false leads altogether, or the objects themselves are just ordinary “things”. This, in my mind, casts a lot of doubt on the underlying premise. While I personally have no doubt whatsoever that there are other forms of life out there, I am extremely sceptical about claims of our skies seemingly swarming with crafts sent by advanced civilisations. Here more than anywhere, the maxim “extraordinary claims require extraordinary evidence” holds. Not only is there no extraordinary evidence, there is in fact no physical evidence at all - just vast amounts of anecdotal evidence, which has no scientific value. So while the above is an interesting and respectful discussion, I think it is really missing the main point. We already know that UFOs are real according to what the term actually means (this is a fact, no need to discuss this), but we can’t know that they are not of terrestrial/natural origin based just on sightings, regardless of exactly how those have been conducted, because there are just too many variables involved.

The entire framework of quantum field theory is built on a spacetime background, usually Minkowski spacetime. You cannot have a quantum field theory of any kind without a spacetime background.

Indeed! Well said.

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.

Thanks, I get you now - you meant that the energy-momentum tensor is the sum of all relevant contributions: \(T_{\mu \nu}:=T_{EM}+T_{matter}+...\) I would like to add to this with the reminder that the Einsteins equations are a local constraint on the metric - so the above is evidently correct in the interior of energy-momentum distributions. But in a vacuum region outside such a distribution, the energy-momentum tensor field identically vanishes - instead, the role of the (now distant) sources is then played by the boundary conditions which one has to impose to obtain a solution to the field equations. What this essentially means is that the Einstein equations alone do not uniquely determine the geometry of spacetime - they only place a local constraint on it. One has to supply extra constraints (i.e. boundary conditions) to obtain a unique geometry. I think this is a really important thing to realise.

\[\sqrt{x}\]

Yes, mass as it appears in - for example - the Schwarzschild metric is a global property of the entire spacetime. Of course. But the mass will still be a global property of that region of spacetime. Sure, that’s an important piece of information. While the internal makeup of the body is irrelevant so long as it remains spherically symmetric, the presence of angular momentum would change the geometry of the surrounding spacetime. They are already taken into account in the total mass of an atom of a given element. Just to add to this, purely to avoid possible confusion - this is the Newtonian picture. In GR, the gravitational fields are not vectors but tensor fields, and they don’t linearly add, meaning that the sum of two solutions (i.e. metrics) to the field equations is not usually itself a valid solution. However, as was quite correctly pointed out, in the weak field regime (such as our solar system), the Newtonian picture is usually adequate, since the nonlinear effects are quite small. Apologies, but I don’t quite understand this comment...? In vacuum outside a central mass, both \(R_{\mu \nu}=0\) and \(T_{\mu \nu}=0\). What do you mean by sum of all fields?

I see you changed over to a new LaTeX system, so just testing this here... \[y(x)=\int xdx \] \[\displaystyle{R_{\mu \nu } -\frac{1}{2} g_{\mu \nu } R=\kappa T_{\mu \nu }}\]

My advice to you would be to stay clear of this type of terminology, most especially in GR, as it will only add more confusion, and does nothing to enhance clarity of understanding. In Newtonian gravity, mass is a straightforward concept, but in GR the question “how much mass is in a given region of spacetime?” becomes very non-trivial and tricky to answer. There are quite a number of concepts of “mass” in GR, and they are all different, and most apply only to specific circumstances. The reason why this is so is because GR is a non-linear theory, which physically means that the gravitational field is self-interacting; energy-momentum is a source of gravity, but so is the gravitational field itself. This means that you cannot really isolate and localise the source that “makes” a given gravitational field; in most circumstances, almost all of it will come from the central body, but there is also some contribution by the field itself. This makes defining concepts such as “gravitational mass” in a precise manner very difficult. It can be done under some circumstances, but you will then need to consider the entire region of spacetime, not just the central body. Again, the answer to this depends on what definition of “gravitational mass” you use, but will be no in most cases. A counterexample would be a scenario where the gravitational system has a quadrupole (or higher multipole) moment, in which case energy-momentum will be dissipated via gravitational radiation. This changes the gravitational field in the immediate vicinity of the system. You cannot just create a massive body out of nowhere, so the question is not physically meaningful. However, if it were possible, then the answer would be yes, the change in the gravitational field would expand outwards at the speed of light. Note that this would be pretty bad news, because a very sudden and large change in gravity would result in an expanding shock front that contains a narrow region of extreme tidal forces. This would potentially be a very destructive phenomenon. No it wouldn’t.

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).

As a silent reader not actively involved in this thread, allow me to offer an observation. Your initial post had one or two points in it that are relevant and worthwhile, but unfortunately your tone and general presence here comes across as arrogant and condescending, You made at least one good point (and some not so good ones), but you failed to communicate them in a proper manner. Please take this as constructive criticism, i.e. an opportunity for growth.

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.