Mordred

Lets get a couple of details out in the open. I'd like to quote a certain section of the link I lasted posted. to quote. In conclusion, we started with a negative energy solution of the Dirac equation and we end up with a solution which has positive energy and reversed 3-momentum. Indeed, if we reverse the momentum 4-vector this is identical with one of the positive energy solutions of Lecture 6. Hence, by reversing the momentum 4-vector of the negative energy solutions we obtain solutions which describe antiparticles with positive energy (physical particles). Another observation is that we started with a negative energy particle solution, which moves backward in time, since p 0 = E < 0 , and we found that this is equivalent to an antiparticle solution which has positive energy and moves forward in time because p 0 = ∣E∣ > 0 . In other words, negative energy particle solutions going backwards in time describe antiparticle solutions which have positive energy and move forward in time. Now think in terms of those particle antiparticles with regards to Hawking radiation. key note the statement " The physical antiparticle is a positive energy solution" you can certainly model as a negative solution but the physical antiparticle will always have positive energy. Hence you need to understand the charge conjugation terms and how that term includes the particle momentum and helicity terms. In QFT for Lorentz invariance the Klein Gordon is used as opposed the Schrodinger. Unfortunately that is the method I most commonly use for anything relating to particle physics. So that is the method I typically use when answering questions relating to anything involving particles. This includes Hawking radiation. As noted the antiparticle has positive energy. Even though literature often refer to the negative energy solutions. Hawking radiation articles also don't typically cover the charge conjugation aspects that identify and define the particle/ antiparticle. In higher grade articles such as arxiv articles your already already to expected to know this detail.

For aid to the OP cosmic time which is the standard for the FLRW metric used to describe our universe is of the form. \[d{s^2}=-{c^2}d{t^2}+a({t^2})[d{r^2}+{S,k}{(r)^2}d\Omega^2]\] \[S\kappa(r)= \begin{cases} R sin(r/R &(k=+1)\\ r &(k=0)\\ R sin(r/R) &(k=-1) \end {cases}\] An important relation is the critical density relation \[\rho_{crit} = \frac{3c^2H^2}{8\pi G}\] the equations that detail the FLRW metric acceleration equation are below. \[H^2=(\frac{\dot{a}}{a})^2=\frac{8 \pi G}{3}\rho+\frac{\Lambda}{3}-\frac{k}{a^2}\] now in the first equation the proper time is the \(-c^2dt^2 \] term above. However in order to understand how that time component is used by the FLRW metric one has to also understand which class of observers are involved and how the cosmic clock is connected to the Hubble flow (commoving time). As opposed to how SR or GR handles it. GR in this metric the time dependence is directly tied to the scale factor a(t). This wiki link actually has a half decent coverage of the time component of the FLRW metric. https://en.wikipedia.org/wiki/Friedmann–Lemaître–Robertson–Walker_metric

Thank you for providing a solid workable reference +1. That article (well the actual dissertation paper corresponding to that article) is used as a baseline test for the cosmological calculator in my signature. (Just an FYI) on that part. There is a couple of important lines here that directly relates to how the time components of the FLRW metric is applied and some considerations with regards to blindly treating the recession velocity under strictly SR transformations. That is denoted in the the opening statement of the article. We can see beyond the Hubble sphere and a strictly SR transformation will give infinite redshift at the Hubble horizon. The Z value provided being z=1.46 roughly (depends on the cosmological parameter dataset used). The article also mentions that standard candles can be used to rule out SR interpretations of redshift. section 4 mentioned on page 6. section 4 being where the details relating to cosmological redshift under GR becomes essential as opposed to SR the article goes on that the equations in 7,8,9 do not accurately describe expansion due to the treatments applied with recession velocity top of page 15. section 4.3 now changes the scenario in how expansion is treated by the following statement. "In this article we have taken proper distance to be the fundamental radial distance measure. Proper distance is the spatial geodesic measured along a hypersurface of constant cosmic time" So time is now treated differently via the proper time defined by that last statement as supported by statement "Time can be treated differently eg correctly calculate recession velocities from observed redshifts . However to do this we would have to sacrifice homogeneity of the universe and the synchronous Proper time of commoving object". lets stop there for the time being ( no pun intended to see how far the OP understands the article and what I just described) Particularly since a huge set of common misunderstandings of how time is treated by the FLRW metric exists most common trying to apply SR directly to recession velocity

while your at it I hope you look up charge conjugation of the photon being -1. With regards to particle/anti particles including neutral particles such as the photon. Here is a reference paper on charge conjugation as it applies to particles and antiparticles. https://alpha.physics.uoi.gr/foudas_public/APP-UoI-2011/Lecture10-Charge-Conjugation.pdf see page 14 for the negative energy solution top of page. That is the QFT treatment applying the creation and annihilation operators in regards to all particle/antiparticle pairs. This article also directly describes the Helicity terms in regards to all particle/antiparticle pairs. Good luck understanding the equation on page 14 without understanding QFT under those groups The group relations outside this article are the U(1) SU(2) groups you need as you will need both the Dirac matrices and the Gamma matrices. Those equations are previously posted including the covariant derivative for each group.

energy density is always positive for every particle in the standard model left hand particles are doublets under weak isospin the singlets are the antiparticles. Left handed helicity is the normal particle. Right handed helicity is the antiparticle. That is in textbooks. Tell me something if the photon wasn't its own anti particle then where does the issue with baryogenesis and leptogenesis in Cosmology come into play ? You would already would have a matter positive universe simply by having photons. So obviously even though the photon is charge neutral it still has charge conjugation. those negative energy states are negative FREQUENCY modes described with helicity. Now lets apply some every day classical physics. What happens when a Negative frequency encounters a positive frequency ? You only need to look at the elastic and inelastic scattering equations to answer that question.

yeah they are typos not pseudoscience. Carrock have you ever bothered to look into charge conjugation with regards to photon/antiphoton yet and look at how Helicity is involved between standard model particles vs their antiparticle ? Can you identify which is the doublet and which is the singlet ? If you have then under QFT the sum of amplitudes given by their cross sections is what determines what occurs in Feymann path integrals. That equation you posted above is literally the photon/antiphoton integrals used in Feymann diagrams. That's the barrage of math I used to provide the needed formulas the key equation though falls back to \[Q+I^3+\frac{\gamma}{2}\] however you have to understand at least U(1) symmetry group mathematics ideally SU(2) as well. Which is what a large bulk of those equations I posted earlier this thread directly apply to

I'm getting tired of accusations from posters that never show any mathematical argument to support their claims. I think I am well within my right that if someone claims I am wrong they are expected to prove just that beyond bland statements. For example those equations you posted Carrock are contained in textbooks on Cosmology with regards to the thermodynamics. Accusing me of pseudoscience when I am describing textbook equations and relations doesn't work in your favor For one thing I have never seen this article before I don't use screen shot reference papers from some lecture for something inclusive in standard textbooks. The reason those equations look familiar is precisely for the detail they are in standard textbooks.

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.