Markus Hanke

17 hours ago, Genady said:

If the metric is twice differentiable everywhere, then its Einstein tensor is everywhere defined

Very good point!

6 hours ago, joigus said:

It shouldn't should it? Einstein seems to have been of the same opinion.

This got me thinking - setting aside singularities for the moment, are there (in a purely mathematical sense) metrics that are not valid solutions to the EFE? IOW, could one write down a metric for which the equations cannot be worked backwards to obtain a corresponding energy-momentum tensor, given we are on a semi-Riemannian manifold?

Again, I’m talking purely mathematically, never mind physical realisability.

I suspect the answer is no.

13 hours ago, AbstractDreamer said:

Thank you all for being really specific and pedantic in your wordings.  I genuinely need this to help understand with more clarity as I know words are a poor substitute for maths.   I will take some time to absorb all this so I can pose questions that make more sense in terms of real physics and mathematics.

Any time

And your questions are good and valid ones, it’s just that they’re kind of difficult to address without lots of maths references. 

The key concept in this is diffeomorphism invariance - that one can describe the same spacetime/geometry in many different coordinates, without affecting any of the physics. Thus, having purely spatial expansion, and a mix of spatial and temporal expansion in the metric, really can be two different descriptions of the exact same physical situation. This is not intuitively obvious, but mathematically rigorous.

16 hours ago, AbstractDreamer said:

But if you had an FLRW universe devoid of energy momentum, it would still be a valid solution for that universe, if energy momentum could exist.  It just has zero value at the start and at least until the moment of observation.

I’m afraid I still don’t really know what you mean by this. The FLRW solution explicitly depends on the energy-momentum tensor being non-zero - an empty universe where it vanishes won’t have the same geometry as FLRW. You can’t get the FLRW solution from the vacuum field equations.

16 hours ago, AbstractDreamer said:

I don't understand how the cause of spacetime expansion is dependent on energy momentum.

The cause isn’t the presence of energy-momentum, since there also exist vacuum solutions that metrically expand (eg the Kasner metric); in my earlier comment I meant that the presence of energy-momentum implies that the spacetime cannot be flat. And by this I meant Riemann flat, which perhaps I should have stated explicitly.

16 hours ago, AbstractDreamer said:

but the actual mechanic of spacetime expansion - dark energy -

Dark energy is also not the cause of metric expansion; it merely influences the rate at which the expansion happens. The expansion itself happens whether there is DE or not.

16 hours ago, AbstractDreamer said:

in a spacetime geometry of net zero energy-momentum (Minkowski spacetime?)  and over a short period of constant scale factor, could you distinguish how much of the redshift in the wavelength of photons is due to spatial expansion and how much is due to temporal expansion?

But in a spacetime that is globally Minkowski, there is no expansion, and thus no redshift. So I struggle to make sense of your question.

15 hours ago, Mordred said:

Unfortunately this is rather misleading to understanding the FLRW metric. The FLRW metric has a curvature term K.

By curvature I meant Riemann curvature - sorry, I should have made that explicit.

2 hours ago, AbstractDreamer said:

Is FLRW spacetime not flat because of curvature caused by mass

It’s not flat because when solving the EFE, you start with a universe filled with energy-momentum, which necessarily has a gravitational effect. 

2 hours ago, AbstractDreamer said:

and the scale factor of expansion that changes over time?

The scale factor arises as part of the solution; it’s not a source term that appears in the initial setup.

2 hours ago, AbstractDreamer said:

If we zero the non-flatness effect of gravity AND zero the non-flatness effect of the scale factor, is FLRW spacetime otherwise flat?

If you remove all gravitational sources, you no longer get an FLRW solution - you’d be in a different spacetime.

On 5/24/2024 at 4:29 PM, AbstractDreamer said:

A flat spacetime metric cannot be obtained by coordinate transformation from an non-flat expanding space-only metric?

Such a coordinate transformation wouldn’t be a diffeomorphism, meaning you are dealing with two different spacetime geometries, and thus two completely different solutions to the EFE. IOW, an expanding-time-only metric would not just be an FLRW spacetime written in different coordinates.

On 5/24/2024 at 4:29 PM, AbstractDreamer said:

How does this refute an expanding time-only metric?

A metric where only the time part is “expanding” describes a completely flat spacetime, only from the perspective of an observer that undergoes some form of accelerated motion. This is not a good description of our universe, since that contains matter and radiation, and thus can’t be a flat spacetime.

On 5/24/2024 at 4:29 PM, AbstractDreamer said:

But if there was a temporal component, could it be observed as distinct from the spatial component?

So far as I can tell (someone correct me if I’m wrong), a metric where both the time and space parts are expanding could be the natural description of an observer who undergoes non-uniformly accelerated motion with respect to the cosmological medium, provided the time part has a suitable mathematical form (if not, it won’t correspond to any physical observer). For such an observer, measurements of both spatial distances and time durations would explicitly depend on when they are performed. He’d see an expanding universe, but also detect proper acceleration, which would play the role of the distinct time component.

On 5/24/2024 at 4:29 PM, AbstractDreamer said:

Let's say one galaxy is only spatially expanding away from us, and the other is both spatially and temporally expanding away from us, and all three locations (two galaxies and the observer) are on a spacetime plane that has observably flat geometry.

The FLRW spacetime is not Riemann-flat, so I’m not sure how to answer this. Given the time component has the right form, the distinction would be in the presence of proper acceleration in the motion of the observer.

On 5/24/2024 at 4:29 PM, AbstractDreamer said:

My question is what might we be missing when we simplify them.   

The details of the spacetime geometry - curvature, geodesic structure etc - are independent of coordinate choices, so I don’t think we’re missing anything.

It’s like having an electric charge - in the rest frame of that charge, you detect only an E-field. An observer who moves past that same charge sees both an E-field and a B-field. In both cases, you have the same electromagnetic field. It’s the same physical situation, seen from different vantage points. But no observer under these circumstances will ever see just a B-field alone, since that would be a physically different situation.

The expanding space situation is similar - you can re-distribute the expansion among the components of the metric by a suitable coordinate transformation (same spacetime, different vantage point), but no observer will ever see a time-only expansion, since that would imply being in a flat spacetime.

4 hours ago, AbstractDreamer said:

If gravity can curve spacetime locally, why must spacetime be flat cosmologically?

FLRW spacetime is not flat, so I don’t quite understand what you’re trying to ask…?

Yes, the Lentz warp drive is potentially interesting. However, as the author of this review paper mentions, unless someone can demonstrate the feasibility of each stage in the entire life cycle - and I would add steerability to this list -, we still don’t know whether this is actually physically realisable or not. 

52 minutes ago, MSC said:

Sometimes makes me wonder if modern technology was a mistake.

I don’t think it was. Life back in hunter-gatherer days would have been much simpler, but also brutal, painful, and generally short. 

There’s nothing wrong per se with technology and civilisation, it’s alleviated a lot of unnecessary suffering, and - broadly speaking - freed up resources that enabled us as a species to pursue things other than immediate survival and procreation, at least potentially. I for one wouldn’t want to go back to the dark times.

The problem is only that one has to have a certain degree of wisdom with it, and that’s where we seem to be lacking. We’ve become completely dependent on our own creations, and in some sense relinquished our freedom to them. That’s problematic, but also inevitable I suppose, since our basic instinctual-psychological patterns have remained the same.

6 hours ago, MSC said:

Go to 1.50 in the video and watch what happens to external time when camera time hits 3.15.20.0000 external time jumps to 99.99.99.9999. What's happening with the clock? 

What you’re referring to is the instant that the camera crosses the event horizon.

The reading on the bottom clock (“External time”) refers to what an external observer who is at rest far away from the BH would see. But the problem is that from such an observer’s point of view, the falling camera never reaches the horizon at all; just before it gets there, it will appear to fall slower and slower, and will visually appear redder and dimmer, until it fades into invisibility. There is no instant on the distant observer’s clock at which the camera is reckoned to have crossed the horizon - the entire region of the horizon and below cannot be mapped into the external observer’s coordinate system at all, because from his point of view such a region cannot be accessed (and vice versa, the falling camera cannot get back out either). Thus, once the horizon is reached, the times on the falling camera’s clock no longer correspond to any times on the external clock, even though the camera continues to fall and continues to accumulate time normally on its own clock. There simply is no longer any meaningful notion of simultaneity at all, whether relativistic or not.

So you see, time on its own in GR is a purely local concept, specific to a specific observer. It may not be shared by others. In order to ensure agreement between observers, you must now use covariant quantities instead.

I came across another good video, concerning in-fall into a Kerr BH (mass, angular momentum). This one contains explanations, and also readings on two reference clocks; I thought some readers here might like this:

17 hours ago, MSC said:

So lets say a stellar mass black hole about 10 times the mass of our sun. Like Gaia-BH1. 3 clocks. One in a safe stable orbit around the black hole, one fixed just before the event horizon, one in freefall.

Let’s further assume that clock (1) is far from the BH, and orbits slowly. This clock will see (2) to be still ticking, but at a much slower rate (compared to itself). It will see (3) to initially fall at increasing speed, while its tick rate gradually slows; as it approaches the horizon its descent will appear to slow more and more, and its tick rate appears slower and slower. Its visual appearance becomes dimmer and redder, and it eventually just fades into invisibility. But it will never be reckoned to have reached the horizon.

(2) sees (1) to be ticking at a much faster rate, and it will visually appear blue-shifted. (3) will appear similar as described above, just at a different rate; it will also never be reckoned to have reached the horizon.

For (3) itself, the time it takes to reach the horizon is finite; the fall time from horizon to singularity is also finite (for a 10 solar mass BH this will be on the order of ~150 microseconds). What tick rates it will see on clocks (1) and (2) will depend on where along its free fall trajectory the clock is - this is a bit of a balancing act between its own position in the gravitational well, and the degree of relative motion between the clocks. However, once it has fallen below the horizon, it should see both (1) and (2) to be ticking faster wrt itself.

17 hours ago, MSC said:

If I drop multiple clocks in free fall, an hour after the other, would the clocks seem to catch up to the first dropped clock, relative to the external stable orbit clock?

After the initial free fall period, they would all be seen to be asymptotically slowing towards the same region of space, while gradually fading from view. But they would never quite catch up.

17 hours ago, MSC said:

See I don't really know how to think about spacetime.

This is not simple, since it’s a mathematical model in four dimensions.

Spacetime is not a substance, so resist the temptation to think about it that way. The best I can offer is to think about it as a network of relationships - it tells you how clocks and rulers at different places and times are related to one another. Each nexus within the network corresponds to a physical event (meaning: a specific point in space at a specific instant in time), and has attached to it an object (called the metric tensor) which, if you tell it a direction, will give you the spacetime distance to its closest neighbouring nexus in that direction. So it’s a network of separations between events. In special relativity that separation is the same wherever and whenever you are (the metric tensor is just a matrix of constants), but in general relativity it may explicitly depend on where and when you are. For events that are more widely separated (not neighbouring), you add up all the individual separations between the nexus that are in between, so you perform an integration.

The global properties of that network as a whole influence local relationships, and the form of local relationships puts constraints on how the global network can look like. Also, if something changes locally, the effects of that will propagate outwards and “ripple along” the network. It’s a bit like a spiderweb.

Be careful about the trampoline analogy - it is just a visual plot that tells you how certain length measurements (rulers) are related along a specific coordinate in a specific geometry. It’s not a complete picture of what this spacetime would look like.

11 hours ago, Genady said:

Nothing. The 4D-spacetime curvature becomes infinite.

Technically speaking, it’s the limit of scalar curvature that diverges - the point r=0 isn’t part of the manifold, so no curvature tensors are definable there. This is why the technical definition uses geodesic incompleteness, and not curvature. That just as an aside

1 hour ago, MSC said:

occams razor suggestion says that it's just simpler for the event horizon to be cloaking some kind of exotic star where the matter density at any single point in the core is finite

The problem with this is that - at least in ordinary GR - once something is below the horizon, a complete collapse is inevitable, irrespective of what that “something” is made of. This is due to the geometry of spacetime itself - below the horizon, the singularity is in the future of every particle that finds itself there. It is thus not possible to have stable, stationary objects there.

1 hour ago, MSC said:

How do you go about even creating theories that don't end in infinites or end of time stuff?

In the classical realm, it is possible to consider alternative theories of gravity that avoid singularities by postulating different types of geometries, such as for example Einstein-Cartan gravity.

In the quantum realm all bets are off right now, since we don’t have a working model of quantum gravity.

1 hour ago, MSC said:

Also is the effect of time dilation upon entering the blackhole exponential?

Time dilation is a relationship between at least two clocks, so you have to specify which clocks you wish to compare.

Generally speaking, the answer depends on what type of black hole you consider (ie the geometry of spacetime), where the clocks are, and how they move.

1 hour ago, MSC said:

be able to turn, look outward and see the sped up universe and the edges of the very black hole the observer is being pulled into evaporating due to hawking radiation? 

Here’s what you would see if you fell into a Schwarzschild BH (only mass):

And here the same thing for a Kerr-Newman BH (massive, rotating, and electrically charged):

As you can see, what you’d observe depends on the specifics of the black hole.

2 hours ago, MSC said:

To the external observer, nothing can reach the inner core.

For a Schwarzschild BH, an external stationary observer would not even see the test particle reach the horizon in finite time (on his own clock).

19 hours ago, DanMP said:

Ok, so why did you offer the equivalence principle as an explanation for the blueshift there, in the twin scenario discussion?

Because the author of that thread explicitly asked about frequency shift, so I offered one particular way to look at that.

Personally though I would altogether avoid any analysis of what happens at every instant in different frames, and simply compare the lengths of the two world lines. That works irrespective of the details of how the twins move, and irrespective of what spacetime they find themselves in. 

10 minutes ago, AbstractDreamer said:

But homogeneity and isotropy in the cosmological principle is an assumption of spatial distribution of energy momentum.

Yes. FLRW spacetime is a “dust solution” - a universe homogeneously and isotropically filled with energy-momentum that interacts only gravitationally. 

10 minutes ago, AbstractDreamer said:

Choosing time coordinates for a solution to EFE such as FLRW, is an assumption of temporal distribution. 

The choice of coordinate system is arbitrary, it represents no physical assumption. You are basically just picking an observer on whose point of view you base your labelling. You can take the ordinary FLRW metric (usually written in what is called Gauss coordinates) and just perform a valid coordinate transformation to arrive at a different point of view; this can be done directly, and has nothing to do with the field equations or the physics. For example, you could choose an observer that is accelerated at all times - you would get a metric that at first glance looks very different, but still describes the same spacetime.

28 minutes ago, AbstractDreamer said:

Right, but FLRW is a particular solution that inherently forbids temporal expansion because of the choice of coordinates.

As I said, the choice of coordinates is arbitrary. For example, if you were to base your coordinate system on a clock that is not comoving with the cosmological medium (eg one that is accelerated at all times, possibly non-uniformly), you would get a metric where both the time and space parts explicitly depend on the t-coordinate. So long as the coordinate transformation is a valid diffeomorphism, this is perfectly allowed, though probably an algebraic nightmare to actually work with.

It’s important again to realise that this describes the same spacetime, just in terms of different coordinates.

What is not possible though is to try and have only the time part expand - there’s no valid transformation that yields this.

37 minutes ago, AbstractDreamer said:

My position is why all the expansion is attributed to spatial expansion and not temporal expansion. 

See above - you could “distribute” the expansion across both time and space parts of the metric by a suitable coordinate transformation, which has no physical consequences. It’s the same spacetime, you’d just label events in it differently. The question is why you would want to do this - it would greatly complicate most calculations relevant to us, since such coordinates wouldn’t straightforwardly correspond to our own clocks and rulers. But of course you can do this, if you really wanted to.

On 4/26/2024 at 9:41 PM, Phi for All said:

We've allowed a few people to accumulate inordinate wealth, more than they can use EVER.

Indeed. The same is true also with power - all the major decisions which affect humanity globally (eg starting wars etc) are made by an extremely small group of individuals. Who has decided that this would be a great system?

15 hours ago, AbstractDreamer said:

So by choosing such time coordinates, it also inherits the assumption that the cosmological medium of time is moving uniformly

Well, the fundamental assumptions underlying this solution are homogeneity and isotropy - if you feed this kind of energy-momentum distribution into the field equations, you get as solution a spacetime that expands. You are free to choose yourself what kind of coordinate system you wish to use to describe this, but obviously it is smart to use a system where your intended calculations are easy.

15 hours ago, AbstractDreamer said:

There is a problem here I cant quite put into words

I understand what you are trying to say. The FLRW metric does rely on the cosmological principle, that’s an assumption we make - that on large scales the universe is homogenous and the same in all directions. Since there’s an observational horizon past which we can’t see, it’s possible at least in principle that perhaps one of these doesn’t actually hold.

15 hours ago, AbstractDreamer said:

But space-expansion IS a theory, as is the more absurd temporal-expansion.   The premise for the theories is from choice of coordinates.

The underlying premise is really the laws of gravity, meaning Einstein’s equations. If you start off with a distribution of energy-momentum that interacts (approx) only gravitationally, then it’s actually difficult to avoid solutions that metrically expand in some way. FLRW is by no means a unique thing, it’s just a particular example of a large number of such solutions. This is not just due to coordinate choices.

15 hours ago, AbstractDreamer said:

Is this saying there is no transformation that will allow only time to expand and not space?

Indeed.

14 hours ago, AbstractDreamer said:

But it seems fundamental to me that if you make a choice, you instantiate something.

And you are correct - you need to pick some boundary conditions to solve the EFE, which in this case is the cosmological principle.

But in GR, the choice of coordinate system has no physical consequences, so it’s not due to that.

14 hours ago, AbstractDreamer said:

If time coordinates are chosen such that earth-bound clocks are comoving with the cosmological medium

The coordinates are chosen such that they correspond to an observer co-moving with the medium; this seems to apply to Earth too, since we don’t observe anything different. We remain in our galaxy, which is part of a local cluster, which co-moves along with everything else.

14 hours ago, AbstractDreamer said:

I'm just saying our sample range of observations is far too narrow to be confident to say our evidence is significant.

In physics you are always restricted by the set of available data - our task is to find a model that best fits this currently available data. If the data set changes, then sometimes the model needs to change too. There’s many possible objections to the Lambda-CDM model, but honestly, right now there’s nothing else that fits all available data better. Let’s just consider this a work in progress. Physics would be boring if all the last words had been spoken already.

To give the twin scenario more “twist”, you can allow spacetime to not be flat (GR twin scenario). For even more twist, allow spacetimes that are topologically non-trivial

15 hours ago, DanMP said:

In this way we can distinguish between uniform acceleration and uniform gravitational field.

As KJW has pointed out, the equivalence principle is a purely local statement, meaning it applies only to small spacetime regions - meaning small volumes over short periods of time. Within such regions, there’s no purely local experiment you can perform to distinguish between the two.

For the twin scenario, the twin can at every individual instant (or short enough time period) be considered to be subject to a uniform gravitational field, where the gravitational potential may vary from instant to instant. This kind of foliation procedure produces the correct results - though I still don’t think it’s necessarily the best way to analyse the twin scenario.

7 hours ago, AbstractDreamer said:

I had suspected that a transformation could orientate it differently such that time does expand with space.

You are of course always free to pick different coordinates to describe the same spacetime - which is one of the central insights in GR. However, when you do this you also change the physical meaning of those coordinates. In the standard FLRW metric, the time coordinate is chosen such that it corresponds to a clock that is co-moving with the cosmological medium, meaning it fits well with our own physical clocks here on Earth, and thus the “phenomenology” of the metric corresponds to what we actually observe, without any need for complicated transformations.

You are free to choose a coordinate system where eg tick rates aren’t constant, but then you need to be very careful how you relate the metric to real-world observations, since the t-coordinate no longer corresponds to Earth-bound clocks.

Ultimately it is best to describe the spacetime in terms of geometric properties that are independent of coordinate choice; in the case of FLRW for example, we can say the spacetime is conformally flat, meaning during free fall angles are preserved, but not volumes.

7 hours ago, AbstractDreamer said:

When you look up cosmological redshift in wiki there is no "Temporal Redshift" type.  That is, redshift caused by an expanding temporal metric.  It doesn't exist.  Not a single reference, no studies, no papers.

These aren’t different “theories”, but simply coordinate choices. You’re describing the same spacetime in different coordinates. KJW has given an example how a “time-only” expansion metric could look like. Ultimately you want to choose coordinates that make your calculations as simple as possible, and that’s often ones based on the cosmological medium. But in principle, the choice is yours, so long as they’re related by valid transformations.

7 hours ago, geordief said:

Does it have or can it be given  a name?

You might want to check the Dictionary of Obscure Sorrows

13 hours ago, DanMP said:

So, here, on the Earth surface, we also see the light coming from stars straight above us "blueshifting" as long as our accelerometer shows that we are "accelerating" upward?

Technically yes, there will be some amount of frequency shift, though in practice the effect is quite small for a weak field such as the Earth’s.

13 hours ago, geordief said:

Do you have any other (or complementary) intuitive ways  of  sitting this process in our pattern of thoughts?

That’s hard to answer, since whether something is considered intuitive or not depends on the person. I kind of like the paths lengths way of looking at it, since most people can relate to it.

8 hours ago, externo said:

Moreover, the path taken by the twin is longer and not shorter, what is shorter is its proper time.

Total accumulated proper time equals the geometric length of the path through spacetime, as I’ve mentioned already. The crucial point is that the two paths are not of equal lengths.

8 hours ago, externo said:

Yet, Wikipedia says that there is a paradox except in the case where we postulate a privileged reference frame.

I think you should read the article more carefully.

8 hours ago, externo said:

How do you know that there is no preferred reference frame where the light is isotropic? You can't know.

I said there’s no rest frame to light, so it makes no sense to speak of “speed relative to waves”. 

8 hours ago, externo said:

There is no gravitational field in an accelerated frame of reference, we are in flat space-time,

There’s no tidal gravity in an accelerated frame, meaning that

\[R_{\mu\nu}=0; W{^{\mu}}{_{\nu\alpha\beta}}=0\]

and therefore

\[R{^{\mu}}{_{\nu\alpha\beta}}=0\]

However, there is a homogenous gravitational field due to proper acceleration locally, according to the equivalence principle. The Riemann tensor vanishes for such homogenous fields, so spacetime remains of course flat as expected. The metric in the accelerating frame, which now contains a term which is equivalent to a gravitational potential, is isomorphic to the Minkowski metric, also as expected.

20 hours ago, externo said:

So you don't have any explanation, you just stick to the math.

The explanation is exactly what the maths says - pick a different path, and you’ll walk a different distance. There’s nothing else to it.

20 hours ago, externo said:

You used Lorentz mathematics, they are the ones who mathematically resolved the paradox.

No, I showed you that there’s no paradox that needs resolving. It seems to me that you’re wilfully refusing to “get” this. 

20 hours ago, externo said:

Because it is he who causes this increase in the signal by changing its speed relative to the waves.

There’s no such thing as “relative to waves”, because light has no rest frame. The speed is always between emitter and receiver.

20 hours ago, externo said:

And why does the turning twin not need to wait for the signal emitted from Earth to propagate to him before observing the blueshift?

Because it’s he who experiences acceleration locally in his frame. The equivalence principle tells us that uniform acceleration is locally equivalent to a uniform gravitational field; differently put, the accelerated twin sits at a different gravitational potential, which implies frequency shift.

21 hours ago, externo said:

If it is physical it implies that the Earth suddenly ages for real during the U-turn

I never mentioned any U-turn - I made it explicitly clear that I made no assumptions about what the path actually looks like, other than it being light-like (thus differentiable everywhere). Why? Because that’s irrelevant, since the difference in clock readings only depends on the total lengths of the two paths. It’s a global measure along the entire journey. Others here have repeatedly pointed this out too. And since both path length and proper acceleration are invariant measures, both twins agree on the outcome.

21 hours ago, externo said:

So math agrees with Lorentz but not with Einstein.

That’s a meaningless statement. I used Einsteinian SR to show how the twin scenario requires no “resolution”.

21 hours ago, externo said:

the acceleration period can be reduced to an infinitely short time

Really? How do you physically realise an instantaneous U-turn with infinite acceleration?

Once again, the clock times are integral measures along the entire journey.

21 hours ago, externo said:

If the the frequency of the electromagnetic signal increases, it is because the one accelerating has changed its velocity in relation to the signal

In relation to the emitter, not the signal. There’s no rest frame for light.

17 hours ago, externo said:

Inertia comes from the ether. When we accelerate the ether manifests itself as inertia.

You didn’t answer my question, you’re just making another unsubstantiated claim.

17 hours ago, externo said:

Instead of the speed of light changing, it is simultaneity changing and the speed of light remaining constant.

It’s invariant, not constant. There’s a difference here. But yes, since the speed of light is finite and the frames are separated, there’s of course relativity of simultaneity; and since acceleration is a change of velocity, the relationship between the frames is time-dependent.

17 hours ago, externo said:

The article clearly establishes that there are physical discontinuities in Minkowski space

I’ve already pointed out that the metric of a relativistically rotating disk is not the Minkowski metric. 

17 hours ago, externo said:

You talk about paths in Minkowski spacetime as if it had a physical reality.

SR is of course “just” a mathematical model, same as any other model in physics. However, the lengths of paths through spacetime correspond to what clocks physically display, so that’s pretty “real” to me.

Like I said, if you connect the same two points in any territory along different paths, it’s hardly a surprise that, in general, these come out at different lengths. What deeper mechanism do you need for that?

17 hours ago, externo said:

You see that the twin ages less over the duration of the entiere journey, which is Lorentz's interpretation.

That’s no one’s ‘interpretation’, it’s what the standard SR maths say, as I have shown you earlier. There are no interpretations involved in this.

17 hours ago, externo said:

As soon as it leaves the Earth's frame of reference it begins to age less, so its perception of symmetry during the inertial journey is false. He believes that the Earth is aging less than him when in reality it is the opposite.

Only one of the frames experiences proper acceleration, which is not a relative measure - both frames agree on who’s accelerated and who isn’t. So there’s not any symmetry in this situation, except during those times when both frames are inertial. But that symmetry concerns the instantaneous rate at which the clocks tick, not their readings - if one of the clocks has first undergone non-inertial motion, and then becomes inertial, their tick rates are symmetrical, but nonetheless one clock displays a different total proper time. The effect of non-inertial motion is accumulative, which is what I pointed out above with line integrals - you have to account for the entire journey.

17 hours ago, externo said:

This is not consistent with Einstein's interpretation that the traveler ages less during interial travels.

I don’t know what you mean by this. 

9 hours ago, externo said:

Redshift is the manifestation of a change in speed between light and the accelerating one.

No it isn’t - frequency shift is due to the fact that energy is a frame-dependent quantity, it has nothing to do with changes in c. Light propagates at the same speed in both frames, but they don’t agree on its energy.

14 hours ago, externo said:

The ether is detectable during acceleration

Really? Please provide references to peer-reviewed experiments that unambiguously (ie not just in your “interpretation”) detect the ether. What is it made of? What are its equations of motion?

14 hours ago, externo said:

Einstein's SR is not capable of processing accelerations, otherwise explain to me what happens during an acceleration according to Einstein. The speed of light remains constant and time changes its simultaneity, is that the explanation?

You really need to stop repeating things that have already been shown to be wrong. You’re not doing yourself any favours.

What exactly do you want explained? One of the frames measures acceleration (using a local accelerometer), and the frames are related by the transformations given in my link, instead of Lorentz transformations.

14 hours ago, externo said:

This article also explains some problems with SR:

This concerns the rotation of rigid objects at relativistic speeds - it’s been known for a long time that this involves a metric that isn’t Minkowski, so that’s hardly a “problem with SR”, but falls outside its scope.

14 hours ago, externo said:

They say that the traveling twin has traveled the greatest spatial distance and the shortest temporal distance.

No. These integrals concern total accumulated proper time; this is an invariant quantity that’s not relative to anything. I deliberately did not use relative quantities, but invariant line integrals.

What the equations say is that (in this sign convention) it is always the inertial clock that accumulates the most proper time between a given pair of events in spacetime. IOW, any clock that doesn’t trace out a geodesic between these events will record less proper time in comparison - and we know of course that there’s only one such geodesic for any given pair of events in Minkowski spacetime.

Therefore, there’s no paradox, and nothing needs resolving. It’s simply that, if you choose two different paths, you can’t in general expect them to be of equal lengths. 

14 hours ago, externo said:

When exactly does he get younger in relation to Earth?

The dilation between clocks is an integral measure - it concerns a comparison between total geometric lengths of world lines, so one must take into account the entire journey. Thus in general you can’t reduce this to a single instant. The most we can say is that the accumulated times begin to diverge the instant the travelling clock ceases to be at rest relative to the Earth-bound clock. Also, he never gets “younger” - he just ages less.

So again - SR very much does resolve this, contrary to your claim.

PS. I remind you again to bear in mind what the twin scenario is fundamentally about - it’s a comparison between total accumulated proper times on two clocks that connect the same two events along different paths. And this is precisely what I mathematically described, not more and not less.