Markus Hanke

Yes, that’s true, I don’t deny that at all. But mathematics as a language is fairly objective, in the sense that - even if you don’t understand any of the underlying physics - you can adopt a “shut up and calculate” approach, and still eventually obtain the correct results. With enough effort and time and repetition this will allow you to eventually figure out the underlying physics. This is of course after you learn the mathematical techniques required - so I agree with you on that point. I don’t think that will happen, based on the fact that it hasn’t happened with any other historical scientist either. For example, Isaac Newton is nearly 400 years in the past now, and in his own time his paradigm was as revolutionary as Einstein’s paradigm in the 20th century, and just as difficult for people of his time to understand. And still, Newton hasn’t been made a deity - on the contrary, his results have slowly been assimilated into people’s basic worldview, and nowadays they are essentially taken for granted, and taught in secondary school; they are now very “mundane”. I see no tendency for him (or anyone else in physics) to be deified. Could you give a concrete example of what you are suggesting actually having happened?

This is neat, but it isn’t really what the OP was referring to. For one thing, the object in question here is an entire strontium atom - not an elementary particle. The other thing of course is that this isn’t a visual image of the actual atom, but merely diffuse re-emitted light, after exciting the two valence electrons of the outer shell with a laser. That’s not the same thing at all. The atom itself has 38 electrons in five shells, none of which is spherical - I wasn’t able to find a good 3D diagram of the orbital configuration, but suffice to say it is pretty non-trivial. So the picture in the link is quite an astonishing feat (kudos to the guy who took it) - but it’s not a “photo of the atom” in the sense I understand the OP to mean.

The problem here is that our sense of ‘seeing’ is a purely classical process - it’s light of certain wavelengths being reflected off macroscopic objects that simultaneously have well defined positions and momenta. But subatomic particles are not classical objects in that same way - so your question is, in some sense, a category mistake; quantum objects don’t ‘look like’ anything, because they don’t obey the classical principles which underlie our visual sense. If anything, you’d have to turn the question around and ask: what would the rest of the universe look like if you were somehow able to piggy-back along on an elementary particle? And I’m afraid I don’t have a good answer for that one. Don’t think of it visually at all - think of it as an abstraction, similar to how an emoji can be an abstraction of someone’s mental state. The essence of an elementary particle is that it is a representation of a set of fundamental symmetries, nothing more. In tech speak: it is an irreducible representation of a symmetry group. So the best and most accurate way to think of elementary particles isn’t as ‘things’ at all, no matter how tempting that may be, but as abstract expressions of symmetry.

I understood what you were trying to say, as this is an area I have been researching extensively myself. Yes, it is possible to generalise MOND into the relativistic domain by introducing additional fields into the GR Lagrangian. Explicit examples are TeVeS (Tensor-Vector-Scalar gravity), GVT (gauge-vector-tensor gravity), STVG (scalar-tensor-vector gravity), various bigravity models, and quite a few others. So as you can see, this has indeed been considered, and a number of models have been developed. But as I pointed out before, all these models have problems of one kind or another - some make very good predictions in some areas, but fail in others; and some can be ruled out on observational grounds. None of these models has been successful enough to really replace the Dark Matter paradigm for now.

Mathematics. But once again, if every generation of physicists was to reinvent the wheel, because they didn’t believe what the generation before them has already discovered and ascertained, then science will never get anywhere. It is of course good to be sceptical and subject ideas to continued testing, but at some point one also has to put some trust into the consensus about what is already been well ascertained through the scientific method.

Yes, it is possible to do this - both in the purely Newtonian domain, and as a relativistic theory. This is essentially what’s known as MOND (and relativistic MOND). The trouble is that the modification yields residual effects even on smaller scales, which can be experimentally tested for; the most well known of these effects would be that in most MOND models gravitational waves would propagate slower than the speed of light; but we know from observation that such waves do indeed seem to propagate at c, which eliminates a large number of MOND theories. The remaining MOND models then cannot fully explain the observed motions of galaxies and galaxy clusters, so they offer no real advantage over traditional Dark Matter.

If the acceleration was large enough, and the rocket able to withstand the forces involved, then you could make this happen in principle, since the air in the cabin is not rigidly connected to the rocket. In practice though it is unlikely that any kind of real-world rocket would survive this kind of acceleration. But regardless, your understanding of the basic principle is correct. No, because acceleration isn’t the same as velocity. A plane may go reasonably fast at cruising altitude, but it takes time to reach that maximum velocity, so the rate of acceleration involved is comparatively small - which is fortunate for the passengers, since otherwise they’d get crushed into bloody puddles during takeoff

Having a teacher makes grasping relativity easier, but it is certainly not a required necessity. Given some familiarity with basic calculus and linear algebra, anyone could read the original paper on SR and eventually figure out the basic principles involved by themselves, though it might take some time and effort. The same is true of GR, though it would be more difficult. The advantage of having a teacher is that we don’t have to do this - others have figured it out before us, so it is easier and much faster to tap into the existing consensus on these matters. Why reinvent the wheel over and over again? But if the case arises that there are doubts about what a teacher says, we can always go back to the original source and check for ourselves. That’s the beauty of math.

I’m afraid I don’t understand what you mean here. GR evidently works very well, in that it makes testable predictions. Belief thus doesn’t come into it.

Not necessarily. Identifying DE with the cosmological constant is only one possible option among several. It is also conceivable that DE is the effective result of the interplay between more than one factor, such as the presence of a cosmological constant in conjunction with some background scalar field. There is no consensus about this as of now. The cosmological constant has orders of magnitude of ~10^-52 per meter squared; for localised solutions on small local patches it is thus irrelevant.

Yes, this would be the best way to look at it. The form of the gravitational field equations is determined by a set of basic mathematical and physical requirements, and the most general form of equation that fulfils these requirements just happens to be the Einstein equations with cosmological constant. It’s essentially just a background curvature that is there even in the absence of all other sources; the presence of such a background curvature modifies all other solutions obtained from the equation - bearing in mind, of course, that this modifications isn’t just a linear combination of solutions, but something more complicated.

You should be able to do this using the equation system you have written down; density and flow are then vector functions of r, and you need to ensure that the proper boundary conditions (at the surface of the mass distribution) are imposed. This will leave you with a system of PDEs along with boundary and initial conditions. You then need to find a suitable numerical algorithm to generate solutions (not my expertise, so can’t help with that), and some suitable way to plot them graphically. All in all this is quite a formidable task, both mathematically and in terms of coding - so best of luck

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

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

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

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

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

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

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

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

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

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

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

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

Precisely. +1