Mordred

Due to the Cosmological principle and the homogeneous and isotropic expansion no point of reference has any preference for showing expansion. It might be easier to understand expansion as a decreasing energy/density. This should lead you to the FLRW metric acceleration equations and the relevant equations of state for radiation, matter and Lambda. The rate of volume change are determined by that equation. In essence the FLRW metric treats the universe as a perfect fluid with adiabatic and isentropic expansion. Should also indicate another piece of evidence of expansion (density changes and CMB blackbody temperature changes.)

Can you show that using the FLRW metric and not relying on a YouTube video. The proper time statement is inclusive in the FLRW line element but one has to recognize that we have different time treatments involved (proper time) commoving time, conformal time and look back time. The common treatment being commoving time to a commoving observer.

A forum label certainly doesn't alter anything career wise nor does it affect any of my physics credentials. Nor will it prevent me from continuing to answer any threads that involve GR/SR. Quite frankly I have never seen Md65336 ever answer any questions involving relativity beyond Minkowskii. For example there hasn't been any effort on his part to help in the PG thread. He questioned my post yet when shown the perturbation tensor being applied doesn't even acknowledge it. As a point of example anyone that knows how the EFE works knows that you have the metric tensor and the perturbation tensor which acts upon the metric tensor. So should have recognized the statement that md65336 posted in this thread as having validity by recognizing how \[h_{\mu\nu}\] Gets applied it's one of the more commonly used tensors in GR treatments and is also used in renormalization procedures. Nor has any error in any mathematics I have ever posted has been shown erroneous by md65336. So the issue is largely how something is verbally described and not how the mathematics itself directly applies.

Whatever you wish to believe quite frankly I've never seen you look at relativity beyond Minkoskii. As such I certainly don't find your opinion of my understanding of relativity as an expert opinion from yourself. As I mentioned I have no interest in defending myself because you don't agree with my understanding of relativity. Have a good nightl

You know so something I'm not going to bother Believe what you like take the Resident expert label for all that it matters of the record one can have expertise in a particular field of physics without being an expert in another.. I' not about to sit here and defend my position over the course of my membership on this forum to you because you disagree with how I understand relativity as opposed to how you understand it. Quite frankly I posted you dozens of references over the course of our discussions in my defense and you typically ignored them. In particular with our discussions on rapidity... In case you haven't figured it out I don't cone to forums to defend my expertise I come to forums to help others. I don't particularly care if you consider me an expert or not.

now were bringing up a different thread ? on a different discussion ? what is your reply to mathematics I just posted ? please link the threads in question I sincerely hope at some point in time you will defend your accusations with some relevant mathematical arguments

correct in the case of the OP on the topic of PG coordinates I would assume he is already is familiar with the relations I just posted. I'm still feeling out how much of the mathematics he has looked at. Particularly to PG coordinates where the line element (worldline ) is as follows \[ds^2=-dT^2+(dr+\sqrt{rs/dT)}^2+r^2(d\theta^2+sin^2\theta\phi^2)\] in order to properly apply this metric those geodesic relations have a different time component than Minkowskii. He will also need the Hubble function a for lifting the coordinate singularity to get to the interior metric. \[r=a(T)\bar{a}\]

The null geodesic aka worldline is a field treatment under GR the interaction is readily described by the perturbation tensor h_{\mu\nu} which acts upon the metric. Lets put the relationship in terms of the metric tensor. \[g_{\mu\nu}{x^i}=\eta_{\mu\nu}+h_{\mu\nu}x^i\] this is the weak field limit as applied to SR as well as the discussion under way where this comes up. For extremely small perturbations described by the equation v<<c the perturbation will only depend on the spatial components. For relativistic you obviously apply the relevant boosts and rotations however the perturbation now applies both the spatial and time components. However in terms of a null geodesics the \(\tau=0\) and \(ds^2=0\) so we can no longer use the time component nor the spatial components. Instead we must use an affine parameter \(x^\mu (p)\) so from the above a slow moving particle \[\frac{dx^i}{d\tau}<<\frac{dx^0}{d\tau}\] i = (1,2,3) the geodesic equation in this case is \[\frac{ds^2x^i}{dt^2}=\frac{c^2}{2}\frac{\partial h_{tt}}{\partial x^i}\] gravitational potential \[\frac{dx^2\vec{x}}{dt^2}=-\vec{\nabla}\phi(\vec{x})\] \[g_{tt}=-1-2\phi /c^2\] coordinate time and proper time then becomes \[d\tau=\sqrt{-g_{tt}}dt=\sqrt{1+2\phi/c^2}dt\] however as mentioned for a null geodesic you require an affine connections described by an affine parameter. I won't go through the full scale solutions via the Christoffels etc those are in textbooks... however applying the affine parameter as well as the Christoffel symbol \(x^\mu (p)\) and Christoffel \(\Gamma\) you get the null geodesic equation that describes the worldline. \[\frac{d^2x^\mu}{dp^2}+\Gamma^\mu_{k\lambda}\frac{dx^k}{dp}\frac{dx^\lambda}{dp}=0\] That is how the perturbation gets applied there is no crackpottery there, as this can be found in nearly every GR textbook in one fashion or another

I think your missing an important detail the four velocity follows the curve Walds General Relativity book has an excellent section detailing that. Section 4.2 if you have a copy. ( if not I can post the relevant relations for you here) One also has to be careful of which synchronization convention is being applied. You have proper velocity so have Lorentz invariance. see section 9.2.1 onward https://www.reed.edu/physics/courses/Physics411/html/page2/files/Lecture.9.pdf PS latex for this site is fairly easy just use \( prior with the same \ and close bracket at the end for inline. for the line latex (autofills the entire line) its \[ same rule on the close bracket ] One thing I do agree on however is that the Minkoswkii metric is pseudo-Euclidean and not Euclidean.

For proper acceleration you would have hyperbolic rotation Do you for some reason feel there is no proper velocity with regards to PG coordinates ? Which if you look above I've shown for. Recall that the gradient term for gravity is under GR/SR a free fall state where the gravity term is the divergence (tidal force) so where is there an issue describing the freefall of particles under PG ? You have no additional force term being applied to cause acceleration. The line element is the freefall null geodesic hence why the setting for angular momentum terms is set at zero.( top of my post.) World lines in Minkowsii space are straight lines where a null (lightlike) vector satisfies the following causal structure four dimensional vector \[v-(ct,x,y,z)=(ct,r)\] \(c^2t^2>r^2\) timelike \(c^2t^2<r^)\) spacelike \(v=c^2t^2=r^2\) null the last condition must apply to be a worldline after all the speed of light is constant for all observers given by c. I assume you already know that spherical coordinates and cylindrical coordinates are still Euclidean though not Cartesian (Cartesian is also Euclidean).

lol sometimes I love the abstract other times not so much. When it comes to GR extremes such as the different metric treatments involved applied at a BH for example Markus is far more familiar with those metrics than I am. I rarely look into them as they aren't particularly applicable for Cosmology or particle physics. I tend to think in terms of Langrangian solutions as opposed to how GR handles spacetime. So I can sometimes give confusing statements such as my earlier post as a result. Simply put I think in terms of the Principle of least action when it comes to geodesics. (including Null geodesics). In the OPs case from what I see of the discussion between the OP and Markus knowing the distinction of how time is handled between the S metric and the PG metric is essential. atm I'm seeing if Mathius Blau has a better coverage of PG coordinates

I assume your aware that PG coordinates uses a different time called PG time correct ? The Schwartzchild metric the relation between coordinate time and proper time is given as follows \[d\tau=\sqrt{1-\frac{2GM}r}dt\] in the above dt becomes infinite at the horizon. PG introduces a new time coordinate to address that issue for notation I will simply use capital T for PG time. so you need to find T in terms of the Schwarzschild metric. Now due to symmetry all directions of space are symmetric so we can only depend on t and r. \[dT=\frac{\partial T}{\partial{t}}dt+\frac{\partial{T}}{\partial{r}}dr\] for an object dropped at rest at infinity in the Schwarzschild coordinates this gives the energy per unit mass as e=1 and the angular momentum \(\ell=0\). \[\frac{dr}{d\tau}=-\sqrt{2GM}{r}\] \[\frac{dt}{d\tau}=(1-\frac{2GM}{r})^{-1}\] gives \[\frac{dr}{dt}=-\sqrt{\frac{2GM}{r}}(1-\frac{2GM}{r})\] however to work out \(\partial T/\partial r\) we need to consider two events that occur at the same time in S time dt=0 but at slightly different radii however PG time is not the same as S time to fall the same distance. The total difference in PG time is \[dT=\sqrt{\frac{r}{2GM}}[1-(1-\frac{2GM}{r})^{-1}]dr\] \[=\sqrt{\frac{r}{2GM}}\frac{2GM}{r-2GM}dr\] \[=-\sqrt{\frac{2GM}{r}}\frac{1}{1-2GM/r}dr\] with \(\partial T/\partial r\) as the rate of change with respect to increasing r we get \[\frac{\partial T}{\partial r}=\sqrt{2GM}{r}\frac{1}{1-2GM/r}\] and the differentials are \[dT=dt+\sqrt{\frac{2GM}{r}}\frac{1}{1-2GM/r}dt\] \[dT=dt-\sqrt{\frac{2GM}{r}}\frac{1}{1-2GM/r}dt\] which is symmetric for infalling and outfalling under change of sign that should highlight the distinction between the time components of the Schwarzschild metric as opposed to the PG coordinates which is a class of solution to remove the coordinate singularity of the EH (thought also not the only solution other coordinate systems do as well)

The null geodesic aka worldline is a field treatment under GR the interaction is readily described by the perturbation tensor h_{\mu\nu} which acts upon the metric. I won't bother with thr rest of your commentary not worth my effort. Not to mention potentially hijacking someone else's thread

Along with what Studiot posted There are numerous reference frames under GR. With regards to the Painleve chart one can utilize the Lenaitre reference frame. https://en.m.wikipedia.org/wiki/Lemaître_coordinates Here is a better article covering Lemaitre frames with regards to Gullstrand- Painleve coordinates https://link.springer.com/article/10.1140/epjc/s10052-023-11370-9

You seem to have a fundamental misunderstanding of the term reference frame. A worldline connects two reference frames. A reference frame can be inertial or non inertial. ALice has one reference frame Bob has his own reference frame. The worldine is the transition between Alice and Bob's reference frames. The choice of coordinate systems does not change this detail due to invariance of coordinate choice. That is a fundamental principle of the Einstein field equations. Ds^2 is the separation distance between the two events Alice and Bob. Ds^2 is not a reference frame but the spacetime path. Every event (observer, emitter ) is it's own reference frame. The coordinate choice doesn't alter that detail

Thankfully I'm in Southern Alberta we get the smoke which affects my wife's asmtha but thankfully my area isn't at risk though I have relatives that my father is putting up in his home from Jasper.

Thanks changed the above

We do tend to get drier Summers due to El Nino. Though they were predicting a drought season as well.

My sympathies for those suffering the loss of their homes and livelyhood in Jasper Alberta. 30 to 50 % loss due to wild fires. https://edmontonjournal.com/news/politics/jasper-townsite-damaged-by-two-wildfires

Dear Algebra please stop asking us to find your X. She is not coming back and don't ask Y !

Make for a good episode of 1000 ways to Die unfortunately that series is no more afiak lol

https://arxiv.org/abs/2004.11140v1 Cl(6) preons handy reference

I've read the paper itself and can attest it's very poorly written with very little usefulness. You can readily tell the author doesn't understand QFT and barely touches on QM. Regardless of peer review or not the paper itself has little to offer that hasn't been examined and tested already. Let's take for example Archimedes spiral vs electron spin. The spin of an electron is a complex number that requires 720 degree rotations to return to its original state. That does not work for the Archimedes spiral with its 360 degree rotation not to mention the detail that particles are not little bullets but under QFT wave excitations. Particle spin is intrinsic it does not have a classical counterpart. A wave also has transverse and longitudinal components not described by any of its mathematics. If anything its equations are rudimentary (easiest to use) of QM. The 17 fields mentioned is incorrect. Any number of fields are includes in any SM model those fields are not restricted to physical (measurable) fields but often are strictly mathematical. There is no exact number of fields of the SM model. The main problem is the acoustic oscillations it describes requires a medium in essence an Eather which we have incredibly high confidence due to Michelson and Morley type experiments of not existing.

Paper is far too lacking for its claims far too much in particle to particle interactions that cannot be accounted for. It barely touches the surface at a QM level. We already have an accounting of the quantum harmonic oscillator between particle to particle interactions those factors are already included in the Feymann path integrals. Nor to mention the Hamilton as it doesn't have any field treatments it doesn't really substantiate its claims. Archemedes spiral wheel with regards to electron seriously lol good luck with that. The other main issue being the Eather qualities to spacetime that would be needed for its oscillations.

The article is based on the Wheeler De-Witt wavefunction of the universe conjecture without using quantum geometrodynamics I will note. Being more a classical examination.