Don't waste your valuable time reading my posts. Lot's of people don't (even some smart ones.).rajnz00 wrote:dubious
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!rajnz00 wrote:Did Einstein say that, or is it just you?
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).rajnz00 wrote:You have made something invalid, valid, by doing what exactly? 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 wrote:You have arrived at that statement by dividing 0 by 0
Faradave » May 8th, 2020, 9:11 pm wrote:Don't waste your valuable time reading my posts.
[exactly zero is] the separation between two adjacent Planck lengths.

davidm » May 7th, 2020, 10:15 pm wrote: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”?
davidm » May 7th, 2020, 10:15 pm wrote:I insist on differentiating between “predetermined” 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. Predeterminism 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.
rajnz00 » May 7th, 2020, 12:01 pm wrote:hyksos » May 6th, 2020, 6:57 pm wrote:.....However, in Specialrel 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
hyksos » 09 May 2020, 05:27 wrote: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 metaclock to describe the dynamics of a changing universe. Metaclocks are different than the translation in dimension four experienced as "time" in an observer's reference frame.
Faradave » 08 May 2020, 22:55 wrote:BurtJordaan wrote: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.
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.phyti wrote:[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'.
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.phyti wrote:The second statement from Taylor & Wheeler would only be true in the frame of a photon, which no anaut will ever achieve.
Yes, and obviously all realizable things have not yet been thought of. That’s what science and philosophy are for.phyti wrote:And a reminder, all things thinkable are not all realizable.
G and EM are both lightlike, thus interact via interval contact. You can view my thread GWiz, my article or my animation. (But beware, this is all resident wizard stuff.)phyti wrote:LIGO detection of gravity waves …Relative to the block universe, where was this event prior
BurtJordaan » May 9th, 2020, 10:05 am wrote:
The central mystery of relativity is: "why is the twoway speed of light in empty space invariant".
BurtJordaan » May 9th, 2020, 12:52 pm wrote:hyksos » 09 May 2020, 05:27 wrote: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 metaclock to describe the dynamics of a changing universe. Metaclocks 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.
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 topdots 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.
hyksos » May 8th, 2020, 11:27 pm wrote:rajnz00 » May 7th, 2020, 12:01 pm wrote:hyksos » May 6th, 2020, 6:57 pm wrote:.....However, in Specialrel 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 onewaystreet
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 metaclock to describe the dynamics of a changing universe. Metaclocks 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.
Faradave » May 8th, 2020, 11:16 pm wrote: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.
davidm wrote: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!
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.

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 onewaystreet
The central mystery of relativity is: "why is the twoway speed of light in empty space invariant".
hyksos » 10 May 2020, 00:42 wrote: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 culdesac.
Feel free to answer the original question.
phyti » 10 May 2020, 20:43 wrote: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.
hyksos wrote: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 onewaystreet
phyti wrote:If changing the sign of a variable alters the behavior of the physical universe, then science should start over.
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.BurtJordaan wrote:That is beside the fact that distance is defined using the average twoway speed of light.
Faradave » 10 May 2020, 23:26 wrote: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.
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.BurtJordaan wrote:is the "interval speed" a valid concept for light? Is "interval speed" a valid concept in at all?

Faradave » 11 May 2020, 19:15 wrote: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
the step ∆d/∆t = √(1  v²) = √(1  v²/c²) = 1/γ
Faradave » May 9th, 2020, 10:20 pm wrote: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 cdependent contact.
Burt Jordaan wrote:Your "interval speed (∆d/∆t)" is just an expression for the relativistic gamma factor inverted. In units where c=1:
Burt Jordaan wrote: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 wrote:1. Does the above mean that a "spacelike" interval has negative spacetime interval separation?
Positor wrote: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 » 12 May 2020, 21:50 wrote: 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.
Fd wrote:interval speed: d/t = √(c²t² – x²)/t = √[(c²t² – x²)/t²] = √[c² – (x²/t²)] = √(c² – v²)
Fd wrote:Zero interval speed from any event accomplishes zero interval separation i.e. contact or colocation in 4D. c is an absolute speed limit because nothing is slower than absolute rest!
Burt Jordaan wrote:The years in 'light years/year' cancel, leaving you with dimensionless 'light'.
Burt Jordaan wrote: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?
Burt Jordaan wrote:"interval speed" has no physical meaning.
"absolute" is often used to mean "invariant".Burt Jordaan wrote:I find [that] totally unintelligible!
BurtJordaan » May 10th, 2020, 11:55 pm wrote:hyksos » 10 May 2020, 00:42 wrote: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 culdesac.
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 tensorform 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/jonathanluninebigbangtheorygeorgeslemaitre
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?
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