The Block Universe and The Flow of Time

[rajnz00]

https://en.wikipedia.org/wiki/Mathematical_fallacy#Division_by_zero

[Faradave]

dubious

Don't waste your valuable time reading my posts. Lot's of people don't (even some smart ones.).

Did Einstein say that, or is it just you?

This is exactly what the moderators are afraid of! Of course it's just me doing the derivation. [see my comment to hyksos] Nevertheless, I stand by it. When I use established cannon, you accuse me of resorting to appeal to authority. Can't win!
And that was some good stuff … colors, diagrams and everything.

You have made something invalid, valid, by doing what exactly? Dividing 0 by 0?

No! For 0/∆t, the zero is notably in the numerator (on top). This isn't a limit either. ∆t is just any inertial observer's measure of the time it takes for light to get from emitter to absorber (detector).

You have arrived at that statement by dividing 0 by 0

No! It's been a long week and you're probably due some R&R. Spacelike and timelike intervals are a difference (subtraction). The name tells you which term is larger. For a spacelike interval (∆d): ∆d²=∆r²-∆t², where ∆r is the interval's spatial component.

[rajnz00]

Don't waste your valuable time reading my posts.

I like reading your posts and my apologies for having got you wrong.

I don't hold it against you for holding your own opinions and deriving original ones.

I can't agree with your conclusions though that there is "direct contact between an emitter (say, 10 billion years ago) and an absorber (say, 10 billion years hence) by light"

Some people get conclusions wrong. (Even smart ones).

I also can't help being sceptical of your handling of zero and "exactly zero" and "contact".

After repeatedly saying exactly zero was contact, you said :

[exactly zero is] the separation between two adjacent Planck lengths.

You can't have separation and contact at the same time.

[Faradave]

Fair enough, rajnz00. There's always a danger of my corrupting innocent bystanders, which is why I usually end up in Personal Theories. It's your thread after all. Enjoy it.

P.S. By "adjacent" Planck lengths, I meant "touching".

[hyksos]

only the indexical present is actual, but all points in spacetime have being. A future temporal a part of me has a “now” that for him is actual; for the temporal part of me that types this, my “now” is actual but my future temporal part’s “now” has being but is not actual. Is this what you are referring to by “possibilism”?

Yes.

I insist on differentiating between “pre-determined” and “fixity” in talking about a block universe. They are, in fact, two different concepts. “Fixity” states that the future is as fixed as the present and the past, but is silent on how it (and the present and the past) gets fixed. Pre-determinism is a possible cause of fixity, and fixity is the effect — which puts the two concepts in different categories. If we don’t know how the block universe gets fixed, as I have argued earlier, then there remain several potentially valid causes for its fixity — freely willed acts among them.

It is difficult to argue that anything gets fixed. The verb, "gets", sounds like temporal change. It is hard to identify anything changing in the Block Universe. This topic was treated here once long ago in another thread. We came up to a professional author who wrote of observers "climbing up their world lines". Much in the same analogy as your description of things "getting fixed" there is likely no climbing either.


.

.

As far as I can see, it is rajnz00 who has recognized that there is no temporal flow in the Block Universe. I mean, raj's physics is atrocious, but his metaphysics is mostly spot on.

[hyksos]

.....However, in Special-rel you still have a difference between forward flow and backward flow. When you get to quantum mechanics, that disappears. The formalism of QM works equally well in either direction of time. ....

I'm pretty sure SR and GR do not have an arrow of time, that their equations work both ways.

Someone will doubtless correct me if I'm wrong

SR has a minus sign on the time coordinate of the metric. This forces time to act differently than space. Some have claimed this is reflecting the aspect of time where it feels like a one-way-street

GR is a different matter. I tried to research the early work of Alexandre Friedman. My motivation was to find out whether he had to include (or even insert) a meta-clock to describe the dynamics of a changing universe. Meta-clocks are different than the translation in dimension four experienced as "time" in an observer's reference frame. After a few hours, I couldn't get anywhere, neither yes or no. My research came up inconclusive.

[BurtJordaan]

I tried to research the early work of Alexandre Friedman. My motivation was to find out whether he had to include (or even insert) a meta-clock to describe the dynamics of a changing universe. Meta-clocks are different than the translation in dimension four experienced as "time" in an observer's reference frame.

Friedmann used Einsteins 1916 field equations to derive the Friedmann equations, i.e.

Friedmann Equations 1922

So the time he used was Einstein's time for a clock sitting at rest relative to the distant stars, far from any gravitational source. He assumed a spatially homogeneous and isotropic universe. This is what today is called 'cosmological time' and represents the time of a comoving clock in free space that 'observes' the CMB temperature as the same in all directions.

The top-dots refer to differentiation w.r.t. cosmological time.

There are many references, but https://en.wikipedia.org/wiki/Friedmann_equations is a good starting place. Many references at the end.

[BurtJordaan]

The inertial frame of the light quantum cannot exist, because then the quanta must still move at c relative to that inertial frame.

This IS a big deal. For SR, Einstein postulated 1. the invariance of physical law and 2. the invariance of speed limit c (as such a law). It's a good thing to have correctly postulated: c=c’. It's a great thing to explain why.

True, but it is important to recognize that it is only the two-way speed of light that is invariant. The one-way speed depends on the synchronization of clocks, which is unique to every inertial frame and hence not a true invariant - it is actually a useful convention in order to simplify physics. It does not matter what method we use to synchronize clocks, be it the Einstein method, slow transport, some astronomical effect or whatever, the frame dependence remain.

The central mystery of relativity is: "why is the two-way speed of light in empty space invariant". I do not quite see where Faradave's model explains that in a non-circular(-argument) way.

[Faradave]

[In the original development of SR by Einstein, the invariant interval was an equality:
x2+y2+z2 =(ct)2 the expression for a sphere. There was no need for a 'null geodesic'.

Sorry for the delay, phyti. The equation you supplied (given the 2's are exponents) happens to define a lightlike interval. True, an expression for a sphere in 3D takes the form: ∆x²+∆y²+∆z² = ∆r², where ∆r is the radius. I'm talking about 4D, where the radius of a sphere would entail all 4 dimensional components e.g. ∆x²+∆y²+∆z²+∆t²= ∆r².
That radius represents the 4D separation between the center and any point on the sphere.

Spacetime however is not Euclidean (it is "pseudoEuclidean", specifically hyperbolic) which has no unit sphere. The separations between 4D locations ("events") are given as spacetime "intervals" (∆d) where: ∆x²+∆y²+∆z²-∆t²= ∆d². The minus sign indicates hyperbolic geometry and the ± allows for both spacelike and timelike intervals. When the spatial and temporal separations are equal the interval separation is zero, defining a "lightlike" interval, which can reasonably be interpreted no other way than as interval contact, albeit c-dependent contact.

The second statement from Taylor & Wheeler would only be true in the frame of a photon, which no anaut will ever achieve.

Actually zero interval separation is invariant, which means it is agreed by all inertial observers. Objects with rest mass can't achieve this, but it is attributed to massless light quanta. Same goes for all electromagnetic and gravitational interactions. This is reality, not abstraction.

My interval-time coordinates reveal zero interval separation when spatial and temporal separation become equal. This is consistent with the accepted interval equation and describes all contact, classical and remote.

And a reminder, all things thinkable are not all realizable.

Yes, and obviously all realizable things have not yet been thought of. That’s what science and philosophy are for.

LIGO detection of gravity waves …Relative to the block universe, where was this event prior

G and EM are both lightlike, thus interact via interval contact. You can view my thread G-Wiz, my article or my animation. (But beware, this is all resident wizard stuff.)

[bangstrom]

The central mystery of relativity is: "why is the two-way speed of light in empty space invariant".

There is no mystery.
The one-way measurement for light measures the relativistic proper-time of the interacting atoms where the atoms at both signal source and sink are in direct quantum contact regardless of their observational separation.

Two clocks are used to measure the one-way speed of light and the clocks are placed at the signal source and receiving ends of the interaction, synchronized to an observer in the center. Two clocks measure the events in their relativistic proper-time as simultaneous because there is no distance between either clock and the event itself.

With the two-way measurement of light, a single clock measures the emission and absorption events, but in this case, there is a measurable distance between the clock and the return signal so there is also a c related measurement of time between the two.

We can measure distance in either units of length or units of time since the two measurements are interchangeable and the ratio will always be equal to c. The ratio c is simply a conversion factor between units of length and time. This is what Olaus Römer discovered in 1676 when he observed that c is a constant ratio between our astronomical measurements for distance and measurements for time.

This makes c a dimensional constant expressed as a ratio and it is not a speed even though it is in units of space divided by time. Our standard units for distance, time, and c are all mutually defined so, given any two, we know the invarient value of the third.

[hyksos]

I tried to research the early work of Alexandre Friedman. My motivation was to find out whether he had to include (or even insert) a meta-clock to describe the dynamics of a changing universe. Meta-clocks are different than the translation in dimension four experienced as "time" in an observer's reference frame.

Friedmann used Einsteins 1916 field equations to derive the Friedmann equations, i.e.

Friedmann Equations.png

So the time he used was Einstein's time for a clock sitting at rest relative to the distant stars, far from any gravitational source. He assumed a spatially homogeneous and isotropic universe. This is what today is called 'cosmological time' and represents the time of a comoving clock in free space that 'observes' the CMB temperature as the same in all directions.

The top-dots refer to differentiation w.r.t. cosmological time.

There are many references, but https://en.wikipedia.org/wiki/Friedmann_equations is a good starting place. Many references at the end.

"The top-dots refer to differentiation w.r.t. cosmological time."
In several other websites I found them parametrizing a-dot and a-double-dot with respect to time.

å(t) , ä(t)

One paper even asserted that d å(t) / dt = H where H is Hubble's Constant.

A parametrized time means the manifold is "changing" in accordance/tandem with the CMB rest frame. When parametrized you cannot freely choose coordinates. You are married to a "clock" parameter. I couldn't tell whether I was reading the original derivation, or whether I was reading a modern treatment of it. I arrived at a cul-de-sac.

Feel free to answer the original question.

[rajnz00]

I'll be signing out tonight, with my observations and conclusions, after this.
But in the meantime I saw this

.....However, in Special-rel you still have a difference between forward flow and backward flow. When you get to quantum mechanics, that disappears. The formalism of QM works equally well in either direction of time. ....

I'm pretty sure SR and GR do not have an arrow of time, that their equations work both ways.

Someone will doubtless correct me if I'm wrong

SR has a minus sign on the time coordinate of the metric. This forces time to act differently than space. Some have claimed this is reflecting the aspect of time where it feels like a one-way-street

GR is a different matter. I tried to research the early work of Alexandre Friedman. My motivation was to find out whether he had to include (or even insert) a meta-clock to describe the dynamics of a changing universe. Meta-clocks are different than the translation in dimension four experienced as "time" in an observer's reference frame. After a few hours, I couldn't get anywhere, neither yes or no. My research came up inconclusive.

https://medium.com/scitech/https-medium ... 9332374d35
"Under Einstein’s 1905 special theory of relativity (“SR”), the Doppler equations have a unique form: If “v” is the recession velocity and “c” is the speed of light, a signal given off from a receding body with energy E arrives with energy

E’/E = SQRT[(c-v)/(c+v)] .

This equation has some unique characteristics. Flipping the velocity sign from positive to negative (to describe an approaching body) flips the equation upside down – as a result, the approach- and recession-shifts for a moving body completely cancel out. The energy you get out is the same as what you put in. This is generally considered to be A Good Thing. Also, if you reverse the velocity-sign AND the energy-ratio (to find the predictions under time-reversal), you end up with exactly the same equation you started with.

So special relativity has no arrow of time."

[rajnz00]

Fair enough, rajnz00. There's always a danger of my corrupting innocent bystanders, which is why I usually end up in Personal Theories. It's your thread after all. Enjoy it.

Well you may have "corrupted" innocent bystanders, charon maybe, or some of the silently watching, but unfortunately you have singularly failed to "corrupt" or convert me.

It's not my thread, I launched it and I will be signing out tonight after giving a few of my thoughts and observations from this thread, leaving it in your and others capable hands.

Science was always used to explain experiences. Why does the sun rise from the East and set in the west? Why do objects fall to the ground?

Science is about good explanations. I read somewhere that a good theory explains a lot with very few assumptions, Newtons Laws, SR, GR, Evolution descent with modification.

It is not sufficient to say some theory, allegedly, says that our experience is an illusion, it must reveal a cause and proof for that illusion, such as the revolution of the Earth about its axis causing the apparent movement of the sun (explanation) and the Focault pendulum (proof).

I have consistently maintained that the present moment, the Now, alone exists, not the past or the future, for every object or individual. Your Now is fleeting, and it recedes into the past.

This Now is unique for every object. My Now is not your Now.

What are my arguments for it? Well, for one thing, it is universally experienced by every individual, like the day following the night following the day. The apparent rising and setting of the sun, from, and into, the Earth may be an illusion, but not the rising and setting itself.

We are continually having experiences (events) in our now. Once they have happened they slip into the past.
Where is our past? We can't travel to it, we can't grasp it.

I can only conclude, with Saint Augustine, that it exists only in our memories. And our memories are short and fleeting.

When we celebrate VE day, or ANZAC day, we say, “Lest we forget”. But forget we do. The older the memory, the less we remember, even collectively. The Universe is designed for wiping old memories and creating fresh ones.

Historians struggle to reconstruct history through the evidence left when the events took place. Buildings, artifacts, graves, skeletons, meteor craters, fossils.

We recreate the history of our Universe by gazing at distant stars, measuring the cosmic microwave background radiation, they are the fossils and skeletons of our Universe.

The future doesn’t exist. Why do I say that? Besides the fact that we can't travel to our future, we can't remember it. The most logical explanation is the future hasn’t happened yet, for each individual or object in the Universe, in their present, or its present Now.

Do I have an explanation for this? Yes. It is my own, I’ll come to that presently.

SR is an elegant theory, thoroughly proven. Does it rule out presentism that I have described above? Not at all.

My arguments are there for anyone to read. I stand by them and don’t retract a thing. But now I will address the arguments raised here why a block universe is the only one compatible with SR is false.

All your arguments rest on one basic assumption that the 'now' in SR that are constructed using surfaces of simultaneity are real and exist, when obviously they don’t! The only Now that exists is the proper time Now of any object.
As phyti so nicely pointed out "World lines do not persist, no more than trajectories. They are histories of positions, and not observable as independent entities. If I want to visualize the Lewis and Clark exploration, there is a map that plots each position over a distance over an interval of time. The image of Tom is NOT Tom!"

I think you denied the block universe model outright[correct] — which Sabine does not do;
indeed, as I pointed out, she believes the block universe is real!

If she does believe that, she doesn’t say so in that post. All she says is that SR is the only model that can resolve the debate on whether the world is three-dimensional or four-dimensional.

But in any case it is irrelevant to our argument.

In fact she specifically argues that presentism is compatible with SR.

She defines presentism as Presentism means there is a notion of 'now' and it is only the 'now' that exists. And she further clarifies that presentism in which time is a factor.

And the argument she gives is basically the same as mine.

"The first observation is that one should be skeptical about this claim [of Pekov's] because it involves the notion of 'existence' which doesn't appear in SR altogether. So how can one possibly say whether some sort of 'existence' is compatible with SR?

So how come Petkov as Rietdijk and Putnam can and do say anything about claim 1)? Well, the reason is that they don't actually explain anything about 'existence'. Instead, they say the 'now' does 'exist', thereby exporting 'existence' into a concept that can be defined in SR"

I note you think that Sabine Hossenfelder is very intelligent, and me not so much. But God hath chosen the foolish things of the world to confound the wise.

Sabine only denies Petkov’s claim that only the block universe is consistent with special relativity. You went further and denied it outright. In order to support you(r) denial and support presentism, I invite you to address my two challenges, above, which I issued to presentists.

I have done so.

Question for the audience, if those who believe in the present are labelled presentists, what do you call those who believe in the Block Universe? I leave it to you to be creative here.

Now to my hypothesis. When I first learned of the Big Bang, I was curious about where the centre was. I imagined it to be like the chrysanthemum explosion of fireworks in the sky. Then I was told that there is no centre, but the “explosion” or expansion takes place everywhere.

Suddenly, I was transported from an insignificant being, on an insignificant planet, of an insignificant star, on the outer reaches of an insignificant galaxy, to the centre of the Universe!

That’s where I am and so are you. We are all individually in the centre of it, with space, and probably us too, expanding all around us, creating space, time and events and our futures as we go along.

That's how our Now comes about.

So that’s it folks. Be of good cheer. Your future is open and it is for you to create with your choices. It has not been decided for you. This is the gift I leave for you. :)

PS Faradave, Lines, points, numbers, zeros are a construct of the mind and dont exist in real life. Pull your head out of your pet theory and look around you.

Sorry being harsh there, but I really mean it about lines and planes and zeros. Please see my first post.

Thank you everyone for participating, engaging and contributing, and enlightening specially to BurtJordaan, davidm, Faradave, bangstrom, hyksos and of course phyti

[Faradave]

rajnz00,

Congratulations on picking such a provocative topic! At over 4,900 views to date your thread blew past every other current science topic including covid-19 (which had a 3-week head start).
It also has nearly 5 times as many posts, some longer (and almost as good) as mine! ;•) Hope to see more topics like it when your time permits.

[phyti]

hyksos;

SR has a minus sign on the time coordinate of the metric. This forces time to act differently than space. Some have claimed this is reflecting the aspect of time where it feels like a one-way-street

Minkowski gets credit for treatment of t using complex notation, since there are only 3 independent variables for spatial dimensions. He then transforms Einstein's equality to a sum of 4 generalized variables. He improves the mathematical methodology for SR but moves the theory farther into the world of abstraction.

If changing the sign of a variable alters the behavior of the physical universe, then science should start over.

[phyti]

BurtJordaan;

The central mystery of relativity is: "why is the two-way speed of light in empty space invariant".

Thought this was answered with this:
The description is that of U, a generic rest frame.
The graphic shows front and back clocks 1 and 2 synchronized to master clock mc, using light (blue) per the SR convention.
Forward motion in the x direction alters spherical symmetry in x direction, replacing it with skew symmetric configuration. The forward path (0, c2, D) rotated 180 deg is equivalent to the rearward path (0, c1, D).
Assume light speed in x direction is c+Δ. Then the transit time from A to c2 decreases , and the transit time from c1 to A also decreases by the same amount, preventing any detection of variation of c.

https://app.box.com/s/095gd0nt79jpvo16com2t92avf0h944s

[BurtJordaan]

One paper even asserted that d å(t) / dt = H where H is Hubble's Constant.

A parametrized time means the manifold is "changing" in accordance/tandem with the CMB rest frame. When parametrized you cannot freely choose coordinates. You are married to a "clock" parameter. I couldn't tell whether I was reading the original derivation, or whether I was reading a modern treatment of it. I arrived at a cul-de-sac.

Feel free to answer the original question.

Yes, då(t) / dt = H was derived by Georges Lemaître around 1927, who found Friedmann's solution independently, but importantly, in the form that is still used today.

But I understand your problem. It is rather impossible to see from Friedmann's original paper "On the Question of the Geometry of Curved Space" how the set of equations that is currently used evolved. However, Georges Lemaître published a paper in 1927 that apparently, contained the tensor-form of the modern equations. I say apparently, because I could not find a publicly available copy of that paper.

I did find a 2017 talk by astronomy professor Jonathan Lunine, director of the Cornell Center for Astrophysics and Planetary Science, where he specifically stated that Lemaître did develop that tensor form in 1927, cosmological constant and all. He said that Lemaître also calculated the value of lambda and the presentation showed extracts of the 1931 English translation of the 1927 French paper. It is around 37 minutes into the talk that I viewed:
https://www.cornell.edu/video/jonathan-lunine-big-bang-theory-georges-lemaitre

The talk in itself is quite interesting and pushed the point that Lemaître should have been credited with predicting Hubble's Law more than a year before Edwin Hubble found it experimentally - and accelerating expansion 70 years before it was observed.

Maybe you have a connection that gives free access to the 1927 paper?

[BurtJordaan]

Forward motion in the x direction alters spherical symmetry in x direction, replacing it with skew symmetric configuration. The forward path (0, c2, D) rotated 180 deg is equivalent to the rearward path (0, c1, D).
Assume light speed in x direction is c+Δ. Then the transit time from A to c2 decreases , and the transit time from c1 to A also decreases by the same amount, preventing any detection of variation of c.

That is true for movement in opposite-parallel directions, but what about two perpendicular directions, like in the Michelson-Morley null result? AFAIK, it requires Lorentz contraction, which depends on the clock-sync method, which again depends on the one-way speed of light. So the usual explanation is circular.

I know some say slow clock transport is sufficient, but it is still debatable on the grounds of sync offsets due to relative movement - which unlike time dilation, is a first order effect of speed. That is beside the fact that distance is defined using the average two-way speed of light.

[Faradave]

SR has a minus sign on the time coordinate of the metric. This forces time to act differently than space. Some have claimed this is reflecting the aspect of time where it feels like a one-way-street

If changing the sign of a variable alters the behavior of the physical universe, then science should start over.

Minkowski's original treatment of coordinates employed convenient ict notation where i = √(-1). But it would have been just as valid (though less convenient) to apply i/c to all three spatial coordinates instead. Now i is generally avoided by sticking with squared terms and letting i become a full minus sign, indicating a hyperbolic geometry. As you know, c is usually also given the convenient value 1 in natural units. So, spacetime interval (∆d) is given by:
∆d² = (∆x² + ∆y² + ∆z²) – ∆t² and just as well by:
∆d² = ∆t² – (∆x² + ∆y² + ∆z²). Common practice is to lead the difference with the larger term(s) and distinguish the associated intervals as "spacelike" or "timelike" respectively. And as I never tire of repeating, lightlike intervals are defined by ∆d = 0 (interval contact).

Bottom line: As phyti suggests, the minus sign does not itself connote anything special like unidirectionality, which is better considered fundamental to time as a dimension.

[Faradave]

That is beside the fact that distance is defined using the average two-way speed of light.

All the more reason to avoid a distance expression for lightspeed. Slope or angle in spacetime might be used but that involves spatial length in the ratio, so I don't think it really helps you.

I prefer interval speed (∆d/∆t) because it becomes invariant precisely when its value is zero (lightspeed). Of course, that makes a round trip a moot point, with zero interval displacement (and speed) in each direction.

[BurtJordaan]

I prefer interval speed (∆d/∆t) because it becomes invariant precisely when its value is zero (lightspeed). Of course, that makes a round trip a moot point, with zero interval displacement (and speed) in each direction.

All the more reason to avoid a distance expression for lightspeed. Slope or angle in spacetime might be used but that involves spatial length in the ratio, so I don't think it really helps you.

Your "interval speed (∆d/∆t)" is just an expression for the relativistic gamma factor inverted. In units where c=1:

The minimum gamma factor is 1 for v=0 and approaches infinity as v approaches 1.

The speed of light is taken as v=c identically and hence gamma must be infinite identically, which is not mathematically allowed.

This begs the question, is the "interval speed" a valid concept for light? Is "interval speed" a valid concept in at all? I think the answer is no, because a spacetime interval is calculated between two fixed events. And fixed events do not have a speed attribute.

So IMO, it cannot be used to as an argument to solve the one-way speed of light issue.

I'm sorry for having gone far off-topic with this thread, so maybe it is time to open another thread dedicated to interval speed.

[Faradave]

is the "interval speed" a valid concept for light? Is "interval speed" a valid concept in at all?

I'm usually happy to employ velocities as fractional light speed v/c. This is actually multiplying by c/c (so we don't even resort natural units), where the numerator shows up when we speak the value (e.g. "0.5c"). This makes writing expressions such as the Lorentz transforms more compact. However, you've run into a problem relating to the very circularity you seek to avoid.

I expressed c in terms of interval speed: ∆d/∆t = 0, but it would be circular to define c in terms of its (fractional) self. Therefore, the step ∆d/∆t = √(1 - v²) = √(1 - v²/c²) = 1/γ must be disallowed.

I appreciate your courage as silence leads me to suspect others suffer the same mental block: geometrically, zero separation means contact, even in 4D.

[TheVat]

While finding many posts fascinating, this reader was left with the sense that the "flow" in the flow of time was never precisely defined. Nor were "moments" more than vaguely outlined (as DaveM pointed out). Nor was the question of the universe being static or dynamically changing really nailed down. Certainly, to our perceptions as biological entities that conceptualize past, present, and future, it seems dynamic. And seems to flow in an unfolding of new events. Our neurological bias is towards free will, because we construe ourselves as active agents of change: new things arise and we feel we can play a role in that. After reading everything here, I still feel that physics is essentially silent on the matter. IOW, physics can never aspire to be metaphysical philosophy, all it can do is measure certain limited domains of energy transfer and particle interaction.

[BurtJordaan]

I expressed c in terms of interval speed: ∆d/∆t = 0, but it would be circular to define c in terms of its (fractional) self. Therefore, the step ∆d/∆t = √(1 - v²) = √(1 - v²/c²) = 1/γ

Quite frankly FD, nothing above makes any sense to me.
And where in what I wrote did you read

the step ∆d/∆t = √(1 - v²) = √(1 - v²/c²) = 1/γ

that you say "must be disallowed"? And how does this couple to the problem of the one-way speed of light?

And where does the final conviction come from?

"... geometrically, zero separation means contact, even in 4D."

Any outside references that is not you own work?

[Positor]

Spacetime however is not Euclidean (it is "pseudoEuclidean", specifically hyperbolic) which has no unit sphere. The separations between 4D locations ("events") are given as spacetime "intervals" (∆d) where: ∆x²+∆y²+∆z²-∆t²= ∆d². The minus sign indicates hyperbolic geometry and the ± allows for both spacelike and timelike intervals. When the spatial and temporal separations are equal the interval separation is zero, defining a "lightlike" interval, which can reasonably be interpreted no other way than as interval contact, albeit c-dependent contact.

Can you please explain the following in simple terms:

1. Does the above mean that a "spacelike" interval has negative spacetime interval separation?

2. Since a zero vector has no specific direction, how can zero 4D spacetime intervals translate to directionally distinct 3D spatial intervals? How can a zero spacetime interval contain the information necessary to specify a particular direction in 3D? If "contact" translates to a particular direction when we move down one dimension, is it really contact?

[Faradave]

Re: c-inside
This relates to my assertion above that interval speed ∆d/∆t becomes invariant precisely at zero, a valid expression for speed limit c as well as explaning its invariance.

Jorrie replied with

Your "interval speed (∆d/∆t)" is just an expression for the relativistic gamma factor inverted. In units where c=1:

Jorrie's interpretation of interval speed.

I attempt to disqualify the derivation below but that in no way impugns the high value I place on Jorrie's criticisms. Physicists get in the occasional computational muddle, only to find they lost track of their c's, which were entered as 1's. Even in units where speed limit c has numerical value 1 (e.g. light years/year), it still has units of length/time which are important when other terms in the equation also employ them.

Simplifications:
Let all action occur along the x axis, so ∆y & ∆z are each zero. I forgo ∆'s by allowing that length, duration and interval are specified by x, t and d respectively.
For example, √(∆x² + ∆y² + ∆z²) is here replaced by x.
t has units of time, x has units of length, c & v have units of length/time.
Roots of a square are assumed ±.

Gamma:
Consistent with Wikipedia, let Einstein's light clock be observed from a primed (') inertial frame to have velocity v to the right . The clock is at rest in an unprimed inertial frame. Two light paths and the clock path (all spatial) describe the sides of a right triangle.

Sides of a right triangle formed by a half path* from Einstein's light clock as seen from two different inertial perspectives.

By Pythagoras: (ct')² = (ct)² + (vt')²
gathering primes: (ct')² – (vt')² = t'²(c² - v²) = (ct)²
dividing both sides by c²: t'²(c² - v²)/c² = t'²(1-v²/c²) = t²
taking the roots: t'√(1- v²/c²) = t
rearranging: t' = γt, where γ = 1/√(1- v²/c²) is unitless (v & c units cancel).

*Wikipedia shows the round trip producing two undeniably congruent triangles (back to back). This demands c be the same for both halves of the trip.

Interval Speed (d/t):
As spacelike trajectories are unattainable (superluminal), interval d will derive from its timelike expression, which can have spatial or temporal units, so long as all components are the same. To keep interval speed (d/t) intuitive, let's use spatial units for d. That's accomplished by using ct for the temporal component in: d² = (ct)² – x².

I now attempt Jorrie's derivation. (My best guess. I'm willing to be corrected.)
timelike expression: d² = (ct)² – x²
interval: d = √(c²t² – x²)
interval speed: d/t = √(c²t² – x²)/t = √[(c²t² – x²)/t²] = √[c² – (x²/t²)] = √(c² – v²)
Here, there is a temptation to convert: √(c² – v²) = √(1 – v²/c²) = 1/γ
But in doing so, the expression loses its units and stops being a speed. It can only be justified by also dividing d/t by c, but then again we're no longer talking about interval speed.

where does the final conviction come from?
"... geometrically, zero separation means contact, even in 4D."
Any outside references that is not you own work?

Zero interval speed from any event accomplishes zero interval separation i.e. contact or co-location in 4D. c is an absolute speed limit because nothing is slower than absolute rest!
That I may be the only one employing interval speed is irrelevant. Physics will catch up …eventually.

[Faradave]

1. Does the above mean that a "spacelike" interval has negative spacetime interval separation?

"...intervals in Minkowski spacetime can be negative."

This is avoided by assigning spacelike intervals a different expression than timelike intervals. The former are associated with superluminal trajectories, which are unachievable. They would theoretically involve travel back in time.

2. Since a zero vector has no specific direction, how can zero 4D spacetime intervals translate to directionally distinct 3D spatial intervals? How can a zero spacetime interval contain the information necessary to specify a particular direction in 3D? If "contact" translates to a particular direction when we move down one dimension, is it really contact?

Vectors have magnitude and direction. A vector can have zero magnitude, yet retain its direction.

Imagine an ice cube, on its east side is another ice cube and on its north side is a hot hors d'oeuvre. Temperature gradient is a vector. To the north of the first ice cube the gradient vector is positive but to the east it may be zero. Both have magnitude and direction. A negative vector is considered in the opposite direction of a positive one.

"Contact" between two geometric points should be considered co-location (i.e. the same point), as Jorrie has noted. But the amount of radial paths leading to that potential contact increase geometrically with dimension.

In 4D spacetime, worldlines contain what we consider speed information in their slopes. It might be argued that a single point can't have a slope but that's exactly what every single point on a sine function has, given as the slope of its tangent line at each such point.

[BurtJordaan]

Physicists get in the occasional computational muddle, only to find they lost track of their [b]c's, which were entered as 1's. Even in units where speed limit c has numerical value 1 (e.g. light years/year), it still has units of length/time which are important when other terms in the equation also employ them.

It rather seems to me that you got yourself into a "units muddle"! The years in 'light years/year' cancel, leaving you with dimensionless 'light'.

interval speed: d/t = √(c²t² – x²)/t = √[(c²t² – x²)/t²] = √[c² – (x²/t²)] = √(c² – v²)

The fallacy of "interval speed" has been pointed out to you many times. A spacetime interval is defined as the spacetime distance between two spacetime events. Event are one-time occurrences and hence are static in spacetime by definition. So "interval speed" has no physical meaning. In each separate inertial frame, an event has a fixed set of coordinates, so where does speed come in?

Some may argue that it comes into play when using space-propertime coordinates, which boils down to plotting "my space versus your time". Here a single event is represented by a vertical line, i.e. a constant spatial, but speed dependent propertime coordinate. This is however just a coordinate dependent construction and does not represent any physical meaning.

Zero interval speed from any event accomplishes zero interval separation i.e. contact or co-location in 4D. c is an absolute speed limit because nothing is slower than absolute rest!

This is the part that I find totally unintelligible!

[Faradave]

The years in 'light years/year' cancel, leaving you with dimensionless 'light'.

Not really (Jorrie?) Light years are distance. Years are duration (a.k.a. time). Light years/year is proportional to meters/second. Now, leave your grandpa's computer alone! He'll be unhappy you played with it while he's asleep.

The fallacy of "interval speed" …A spacetime interval is defined as the spacetime distance between two spacetime events. Event …static in spacetime by definition. …In each separate inertial frame, an event has a fixed set of coordinates, so where does speed come in?

Intervals (d) are invariant, agreed by all inertial observers because they are the same in every inertial frame. Duration (t) is relative, specific to each inertial frame (e.g. their measured durations for light from an emitter to its absorber). Interval speed (d/t) is therefore not inherently invariant. It is generally as relative as conventional speed (x/t). However, a unique exception occurs when d = 0 (defining a lightlike interval). Then all inertial observers agree that d/t is zero. Thus, the interval speed of light is invariant.

"interval speed" has no physical meaning.

We seem to agree interval speed: d/t = √(c²t² – x²)/t = √[c² – (x²/t²)] = √(c² – v²).
√(c² – v²) has units of speed, such as meters/second. For light quanta v = c, consistent with interval speed zero. And all inertial observers will find their relative x and t have the invariant ratio c for light.

Physically in 4D, the emission event and the absorption event are the same single event (despite differing spacetime coordinates). Light travels at zero interval speed going nowhere (interval contact). Light's worldline has zero interval span, but projects various spatial (and temporal) spans in different inertial frames.

Don't you think it odd that Einstein's light clock experiment ("gedanken") gives two equally-legitimate but different spatial paths for the same ray of light.* That tells us that neither is the "real" light path (just two projections). Relativity brings us to understand invariance as being most real. I give you invariant speed and invariant contact. That's the physics behind their physical meaning.

c is an absolute speed limit because nothing is slower than absolute rest!

I find [that] totally unintelligible!

"absolute" is often used to mean "invariant".
"absolute speed limit" refers to invariant speed limit c.
"absolute rest" means invariant interval speed zero (d/t = 0).
I equate invariant speed limit c to invariant interval speed zero.

It's fine to disagree but I hope that's clearer.

*Grandkids, this is NOT Young's double-slit experiment.

[hyksos]

One paper even asserted that d å(t) / dt = H where H is Hubble's Constant.

A parametrized time means the manifold is "changing" in accordance/tandem with the CMB rest frame. When parametrized you cannot freely choose coordinates. You are married to a "clock" parameter. I couldn't tell whether I was reading the original derivation, or whether I was reading a modern treatment of it. I arrived at a cul-de-sac.

Feel free to answer the original question.

Yes, då(t) / dt = H was derived by Georges Lemaître around 1927, who found Friedmann's solution independently, but importantly, in the form that is still used today.

But I understand your problem. It is rather impossible to see from Friedmann's original paper "On the Question of the Geometry of Curved Space" how the set of equations that is currently used evolved. However, Georges Lemaître published a paper in 1927 that apparently, contained the tensor-form of the modern equations. I say apparently, because I could not find a publicly available copy of that paper.

I did find a 2017 talk by astronomy professor Jonathan Lunine, director of the Cornell Center for Astrophysics and Planetary Science, where he specifically stated that Lemaître did develop that tensor form in 1927, cosmological constant and all. He said that Lemaître also calculated the value of lambda and the presentation showed extracts of the 1931 English translation of the 1927 French paper. It is around 37 minutes into the talk that I viewed:
https://www.cornell.edu/video/jonathan-lunine-big-bang-theory-georges-lemaitre

The talk in itself is quite interesting and pushed the point that Lemaître should have been credited with predicting Hubble's Law more than a year before Edwin Hubble found it experimentally - and accelerating expansion 70 years before it was observed.

Maybe you have a connection that gives free access to the 1927 paper?

I don't have it.

However, this is where I ended up.

https://opencommons.uconn.edu/cgi/viewcontent.cgi?article=1464&context=srhonors_theses

Relevant material appears on page 16. They leave out the parameters on a for brevity/clarity. a=a(t) But they take the derivative w.r.t. t , meaning that å must be parameterized by t .