Markus Hanke

It should be noted here also that in curved space-times, conservation of energy-momentum is a purely local conservation law; it does not necessarily apply globally across regions of non-Minkowski geometry, unless the spacetime in question has the specific symmetries that give rise to such conservation laws.

“Rest” is indeed frame dependent, but proper acceleration (as opposed to coordinate acceleration) is not - it’s something that all observers agree on. I think you may have misunderstood what I was attempting to say, possibly I didn’t make my thoughts clear enough. Of course the principle always applies - it is a fundamental principle of physics, derived from the principle of least action, and there are no exceptions to its validity in the classical world. However, calculating a specific free-fall geodesic is an operation that must depend on initial and boundary conditions, since the underlying equations are differential equations. You cannot obtain a specific, unique solution to a differential equation without imposing some form of initial/boundary condition. So this is how different specific geodesics of the same spacetime are distinguished - they are just different solutions to the same equation (i.e. the same principle of extremal ageing). So you can have different free fall world lines around the same central mass, all of which naturally adhere to the principle of extremal ageing; but they also represent different sets of physical boundary conditions, which you’d need in order to physically realise such trajectories. So the uniqueness of solutions to the geodesic equation is preserved. Hence, the geometry of spacetime along with the principle of extremal ageing determine its geodesic structure, and boundary conditions pick out the specific geodesic that is being realised. Does this makes more sense now?

Indeed, you are right, I didn’t quite think this detail through. What’s more, if we want to be dealing with free fall world lines, then the clocks can’t really be at relative rest at the final event, as this would imply acceleration.

That is not the definition I would use, since the possible paths connecting events depend on initial and boundary conditions as well (specifically - whether your test particle starts off at rest, or has initial momentum). It is better to say that the principle implies that any free-fall path connecting two given events must be a geodesic of spacetime. In singly-connected spacetimes, this choice is unique, since for a given set of appropriate boundary conditions, the geodesic equation has precisely one unique solution. Technically you could have spacetimes the topology of which is multiply-connected; in those cases you can then have more than one geodesic connecting the same two events. I don’t know though if it would be possible to have more than one geodesic of the same length connecting the same events. That is an interesting question, but I’d have to think about that first before attempting an answer. I certainly couldn’t think of a physically realisable example right now. Well, the technical definition is that you start with the integral I quoted earlier, hold the start and end points as fixed, and then vary the paths between these points. The principle simply states that the path physically taken is the one that is an extremum of this proper time functional, i.e. the longest one. Note that the principle of extremal ageing is a specific example of a more fundamental principle, the principle of least action. That is the fundamental principle that underlies all field theory frameworks, specifically quantum field theory, and hence the Standard Model. Also GR itself follows from it - applying the principle of least action to the Einstein-Hilbert action gives you the Einstein field equations.

I think it is really important to emphasise that in spacetime, you can only meaningfully compare clocks that start together at rest, and end together at rest, i.e. which connect the same two events. This is what the principle of extremal ageing applies to - if you have a number of different paths which connect the same two events, the principle tells us that that path which represents the longest proper time (i.e. the greatest geometric interval) is physically realised. Schwarzschild spacetime admits a number of Killing vector fields (which is to say it has certain symmetries), so you can extend this principle a little bit in order to compare setups like the one in question here. I think the relevant scenario would be if you have two events A (near a central mass) and B (further from a central mass) which are situated along a radial line with the centre of the mass. How would a test particle move - will it fall from B to A, or would it go from A to B? If you do the maths, you’ll find that - even though the purely spatial distance is the same in both cases -, the world lines A-B and B-A nonetheless have different geometric lengths in spacetime. The length of B-A is longer than the one of A-B, so a clock freely falling from B-A records more proper time than a clock riding a rocket propelling itself from A-B. This is because the path A-B involves proper acceleration in the frame of the clock, which always implies extra time dilation (ref equivalence principle) on top of the background curvature of spacetime; the free-fall path B-A does not. So B-A will accumulate more proper time. Technically speaking, the curve B-A is a geodesic (segment) of Schwarzschild spacetime, whereas A-B isn’t. The presence of proper acceleration always means a shortening of a test particle’s world line. So to make a long story short - test particles tend to remain in free fall, because that is how they accumulate the most proper time on their clocks => principle of extremal ageing. Any external force implies local proper acceleration, which in turn implies time dilation, leading to less proper time recorded between the same set of events. This is why test particles which start off at rest always fall towards a central mass, never away from it - because the two world lines aren’t the same ones, even though the spatial trajectory may appear to be. Obviously, if you start off not at rest, but with some initial momentum, much more complicated dynamics may result. Intuition quickly fails for such scenarios, and it’s really a matter of doing the maths then.

The total proper time accumulated on a clock between two events in spacetime is equivalent to the geometric length of the clock’s world line C that connects these events, hence: \[\tau=\int _{C} ds=\int _{C}\sqrt{g_{\mu \nu } dx^{\mu } dx^{\nu }}\] Just to clarify - I mentioned the comparison to a far away clock only as a pedagogical aid to illustrate a principle; such a comparison is of course not physically precise, because these two clocks would connect a different set of events. Come to think of it, I should not have brought this up at all, as it only confuses the issue further. Please consider it retracted. You can’t really compare such clocks, since there is no good synchronisation convention you could use for this. Not necessarily, because the total proper time generally depends on the path that is taken as well as the metric (see line integral above). For example, in an equatorial orbit around a rotating black hole, you would get different readings depending on whether your orbit the black hole in the direction of its rotation, or in the opposite direction. Apologies, I don’t quite understand what you mean with this...? Possibly they could age the same, but not necessarily - again, because the total time accumulated depends on the metric and the path that is taken in spacetime. So it really depends on how they travel, and what the geometry of the underlying spacetime is like. There is really no generally valid answer to this, it depends on the specific case. I am personally finding that, even though my intuition about all things GR has gotten far more accurate over the years, it is still a slippery slope and sometimes leads me to conclusions that later turn out to be wholly wrong. In cases like the above, really the only way to be sure is to do the math, and evaluate the above integral for the specific case at hand. In very general terms, we can say that the longest possible path through spacetime between two fixed events is always a geodesic of that spacetime. You can take the above integral, fix the start and end points (i.e. decide on two fixed events), and then vary it between all possible paths between these events, finding the extremum - what you will find is that you end up with the geodesic equation. Because this is a differential equation, its solution is determined by initial and boundary conditions - so if you have a test body coming in at high speed, you might get a free fall geodesic that is entirely different from the one you’d get if you start off at rest. The dynamics are really quite complex, especially in non-trivial spacetimes, so intuition often fails here. It’s best to just sit down and calculate. For your scenario with highly elliptic orbits, it is also very important to remember that orbital distance and orbital speed are not good indicators of how long the world line of the test particle is in spacetime. You definitely have to do the maths here to tell the whole story.

Proper time is the geometric length of a test particle’s world line in spacetime - not just in space, and not just in time, but in spacetime. What we find is that in spacetime, the longest possible trajectory between two events is always a geodesic, which is a world line where the test particle does not feel any acceleration at any point. If you start off near a massive object, the only trajectories that avoid you feeling proper acceleration in your own frame are generally those that bring you closer to the massive object (free fall, in the classic sense), since you would need acceleration to do anything else (unless of course you already come in very fast, e.g. in a slingshot manoeuvre). So think about it geometrically - very loosely speaking, world lines near a massive object tend to be longer as compared to similar ones far away, because spacetime there is “stretched out”, particularly in the time direction (that is why e.g. a radar signal passing by a massive planet takes longer to get to its receiver - because its world line through spacetime is longer). Or you can think of it this way - if you sat on a clock somewhere near a massive object, and look back at another reference clock that is somewhere far away, then the far-away clock would appear to go faster. This is due to gravitational time dilation between your own frame, and the far-away frame. So actually, more time is accumulated (very loosely speaking) near a massive object, as compared to anywhere else. Note that this can either be a maximum or a minimum, depending on how you choose the signs in your metric. That’s why it is more generally called the principle of extremal ageing.

A tensor is a multilinear map that maps vectors and 1-forms into other objects. This is true even if the manifold isn’t endowed with a metric, and regardless of the number of dimensions. The vector cross product which you are using here is defined only in 3 dimensions, and is not generally covariant. It also requires the presence of a metric. Furthermore, if you replace a higher rank tensor by a real number, you loose both information and a number of degrees of freedom. Again, you cannot simply replace replace a rank-n tensor field by a scalar field; they are not equivalent objects. Physically speaking, you need to be able to capture all relevant symmetries of the field (in the case of GR, diffeomorphism invariance), which is possible only if it is a rank-2 tensor field. Remember also that, in terms of QFT, the graviton needs to be a massless spin-2 boson, which again means you need a rank-2 tensor field. It appears that you chose your “characteristics” in order to obtain the solution you wanted, not because those choices are motivated mathematically or physically; there is no mention of any boundary conditions. Vanishing Christoffel symbols imply a flat region. Clearly, this would not lead you to the Schwarzschild metric, or any spacetime other than the trivial Minkowski one. I’m afraid it is not. Also, the fact that the EFE is a rank-2 tensor equation has deeper reasons; you can’t just replace it with a scalar relation. No, mostly because these are not generally covariant objects (no diffeomorphism invariance), and do not capture the underlying physics.

As I’ve pointed out earlier, there is no such thing as a “magnetic charge”.

There are no magnetic charges; magnetic field “lines” either extend to infinity, or form closed loops.

We know which options exist, given the validity of the laws governing gravitation (this can be worked out analytically). We then take the observational data available to us at a given point in time, and see which of these options the data fits best; this then becomes the current consensus. But this is an evolving process - as new data becomes available to us, the consensus can shift over time, as the larger data set may better fit a different option. This is the whole point of physics - it makes models which describe aspects of the universe as accurately as possible, but it does not pursue some spurious notion of “absolute truth”. A model is “true” only in the sense that it fits available data, and makes accurate predictions. As such, it is really more epistemic than ontic in nature.

The (1,3)-dimensionality of spacetime has a fairly privileged character, you can’t just add or take away macroscopic dimensions, and still expect everything to work as before. Adding an extra macroscopic spatial dimension would be really bad news - there would no longer be any inverse square laws; gravitational orbits would be unstable; electromagnetism would no longer be described by Maxwell’s laws; atomic orbitals would look very different or not exist at all; and so on.

Just to add to this - lack of energy isn’t really the reason why no escape is possible. Even if - hypothetically - you had an infinite amount of energy available to you, there still wouldn’t be any way out. Below the horizon of a Schwarzschild black hole, the geometry of spacetime is such that all future-directed word lines of test particles must inevitably terminate at the singularity (in the classical picture of GR), because space and time are related in such a way that ageing into the future always implies an in-falling in the radial direction. This is due to the geometry of spacetime itself, not lack of acceleration from your thrusters. The best you could do is to prolong your inevitable fate by firing your thrusters really hard, which slows down the radial in-fall, but cannot stop it - even remaining stationary at some r=const is not possible. The only way to escape such a black hole would be to time-travel backwards into the past, which, to the best of our current knowledge, is not physically possible.

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

The speed of light is an invariant, it is the same everywhere locally. What you will observe however is that, if you send a beam of light through an extended region of curved spacetime, it will experience a frequency shift (ref Pound-Rebka experiment). I’ve said this on another thread recently - you need to remember that relativistic effects are relationships between frames at different places and times in spacetime, they are not physical changes that somehow happen to the local frames themselves.

Indeed, even though not all 3-body systems evolve chaotically. There are special cases when you get regular, periodic solutions; also, if the mass of one of the bodies is very much smaller than that of the other two, the setup effectively reduces to a 2-body problem. So it depends on the circumstances. But you are right, most gravitational 3-body systems evolve chaotically.

Cigarette smoke rising in air is an example of an open system, the dynamics of which are chaotic. That means we understand the underlying laws very well, and the system is completely deterministic and classic, yet it is still not predictable into the indefinite future. That is because any future state of the system is highly dependent on initial and boundary conditions, and initially small inaccuracies in determining those conditions compound very quickly, making final outcomes increasingly unpredictable the further you go into the future. A satellite orbit is different, since such a system is not chaotic in nature; the orbit of course still depends on initial conditions, but once it has been entered, it will remain stable and not evolve in unpredictable ways. Scale is also a factor - any inaccuracies in initial condition will generally be negligibly small when compared with the total orbital radius of a satellite, whereas even the tiniest movement of air will have a large impact on cigarette smoke. To make a long story short, the differences are not in our understanding of the underlying laws, but in the dynamics of how a system evolves, specifically in the way those dynamics depend on initial and boundary conditions.

This is not correct. The other planets are subject to relativistic perihelion precession as well, it’s just that the magnitude of the effect decreases as you move further away from the sun, so it is more difficult to detect. But the effect is still real even for the other planets, and Newtonian gravity fails to model it correctly.

I consider myself an external observer to this, since I am not a citizen of either the US nor Iran, and have little interest or emotional investment in politics. It is striking to me that no one mentions the ethical dimension of this. This person - regardless of what he may or may not have done - was a human being, and as such has the basic right to life, and to a fair trial, as we all do. How can it possibly be ethically acceptable that anyone - regardless of his rank or position - can order the blatant killing of another human being, just like that? What Trump has done is akin to me hiring a contract killer to get rid of someone I don’t like; if I did that, I would be brought before a judge, and locked away for life, quite regardless of who the deceased person was or what he/she has done. And quite rightly so. Why do these same standards of law, justice, and human rights not apply to the President of the US? Why do they not apply to other heads of states, who commit similar crimes against humanity? At the very least, this Iranian general would have been entitled to a fair trial before an impartial judge. What happened here was premeditated murder, plain and simple. If this act was illegal under US law, then Trump needs to be held accountable for his action to the full extent of the law, like any other citizen would; if it was legal under US law, then that means the US has deteriorated to a point were human rights count for nothing, and lives are expendable for political gain and leverage. If someone is in the way, just have him killed, no need to bother with fair trials. Just to be clear, this is not exclusive to the US, it applies to anyone who acts with impunity in the face of basic human rights. If politics have become more important than life, then humanity is in a bad state indeed.

Some of the interactions I had with them were of quite a threatening nature; it had a really sinister feel to it, far beyond mere online crackpottery. These people are like the Westboro Baptist Church of the science world. Really sad to see.

I was unfortunate enough to have had dealings with members of that organization on another science forum some years ago. There is no reasoning with these people. Your best bet will be to not engage at all - immediately block all attempts at private communication/spam, and keep reporting public reviews, messages, and posts to the respective platform provider, such as Amazon. There’s little else you can do, I’m afraid.

Though I don’t feel strongly enough about the issue to motivate me into spending time to fact-check these claims, I am immediately suspicious of this article. Some of the substances mentioned here would - to the best of my limited knowledge - not even have been available during Einstein’s and Tesla’s lifetimes, or were not known back then to have hallucinogenic effects. I’m tempted to call BS on this, though I’m open to be told differently by someone more knowledgeable in matters of history.

Good point! I did not think of this one. Suppose you have two clocks - you keep one down on Earth, and put the other one into a geostationary orbit. Newton works really well in allowing you to calculate all you need to know about the orbit of the clock in space, as well as the physics of the rocket launch to get it there. However, you will empirically find that, when you compare the readings of the two clocks, they actually record (slightly) different intervals; Newton fails to explain or predict this. The differences in readings will be small, but they still matter and need to be accounted for in some real-world applications, such as e.g. GPS satellites. This phenomenon is called gravitational time dilation. This is just one specific example, there are many many others.

Even if that were the case, you would still see relativistic effects - for example gravitational time dilation, frame dragging around rotating bodies, gravitational light deflection, relativity of simultaneity etc etc. It would not really change anything. The theory of General Relativity, in this particular context, is largely about the empirical observation that (small enough) freely falling bodies are not subject to any forces - which is to say that if you attach an accelerometer to such a body, the instrument will read exactly zero at all times during free fall. That’s why you get a funny feeling in your tummy as you fall after jumping off a high board into a swimming pool. And yet free fall bodies are very clearly affected by gravity, so the force-based Newtonian model (though it works very well as an approximation) is ultimately not adequate as an explanation for gravity. Also, when you start looking at situations involving very strong gravitational fields, the Newtonian model makes increasingly inaccurate numerical predictions. This would continue to be the case, even if everything were nicely spherical and regular. P.S. Even Newtonian gravity does not predict everything to be spherical and “synchronous” - depending on the particulars of a given situation, you can get some very complicated dynamics in Newtonian gravity, too. You can even get chaotic dynamics, i.e. situations that are deterministic, but still not predictable into the indefinite future. So even simple laws can lead to complex outcomes.

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.