Mordred

@Markus Hanke I've been trying to find a decent method to explain geodesic congruence in regards to PG and Lemaitre frames without getting too technical. Have you perchance come across any decent treatments. Blau and Wald may be a bit too intense

Ok with PG coordinates there is some very different treatments from the Schwarzschild Metric the first is the use of the raindrops. Each raindrop has its own clock falling in from infinity. The other feature is that for the lifting operation (singularity treatment) the time coordinate is tied to the scale factor a in a similar fashion as a commoving observer for the FLRW metric (but don't confuse the two). The light-cones are also defined differently in This case the tilting has a different cause. That cause is due to the above two conditions which lead to a different light-cone definition. This is where a reference you have been using would be helpful If you don't have a reliable reference either Markus or myself can likely find one for you. However it may be best to use a reference that you find easier to understand and relate to. Recall earlier I linked the following https://en.m.wikipedia.org/wiki/Lemaître_coordinates this formula is not included in that link but arises from the raindrop coordinates using the Symmetries mentioned in the above link \[dT=\frac{\partial T}{\partial{t}}dt+\frac{\partial{T}}{\partial{r}}dr\] I'm going to drop this reference as it certainly will lead to unnecessary confusion particularly since you already stated afterward that you are not strong in GR mathematics. That and were not really concerned with Kruskal diagrams at this stage https://link.springer.com/article/10.1140/epjc/s10052-023-11370-9

Lol truthfully wasn't worried on that score to begin with not that it matters in the slightest. As far as multiverses go, each universe can indeed have its own mass/energy distribution leading to different expansion rates and critical densities. However our region of causality will be limited to our universe unless some collision occurs. There is a line of research that looked directly for signatures in the CMB and the axis of evil from the first Planck dataset was a contented possible signature at that time prior to realizing that the dipole anistrophy was due to unaccounted local effects from our peculiar motion. Calibi-Yau doesn't particularly deal with multiverse but you do see papers suggesting such much like you do in the many worlds interpretations of QM. In all honesty though I myself focus on this universe as multiverse theories are far too speculative Just a side note though our Observable universe is simply our region of shared causality. When runaway inflation in early inflationary theories was determined attempts were made to solve runaway inflation. One of those attempts being Chaotic eternal inflation which leads to bubble universes (different regions with different expansion rates ). Sometimes coined Hubble Bubble universes. One can however readily determine shared causality regions such as is done for our Observable universe (Cosmic event Horizon) It's been a long time since I saw the movie but there was a lot in that movie that wasn't very realistic when it came to that universe. If I recall even in space they had an atmosphere such an maas/energy density would have likely collapsed but hey it's Sci fi

Lol +1

Yes but look back on the T relations I provided back on page 1 the links to that site never seem to direct me to where it should be going and the site being in French makes figuring out why I'm getting redirected isn't helping. Been kind of busy atm hopefully this eve or this weekend I will have time to assist on the Time components specific to PG as it uses its own time for proper time T and replaces \(\tau\)

T being the PG time it might be best to agree on a specific article at this point as nomenclature between treatments can lead to confusion. We have to look directly at the covectors and vectors at this stage. Covectors are needed in curved spacetimes whereas in Euclidean space vectors are sufficient.

Now your getting it recall \[ x^\mu\] is the particle. X is always the coordinate axis for the particles momentum. In the Minkowskii case the changes to x is only the rescaling for length contraction. In curved spacetime however the x axis is changing orientation as it's the tangent vector on the hypershere. That should be a huge help to making the mathematics of GR easier to understand +1

Hiding behind authority meaning recognizing and using GR correctly. The problem here is you keep repeating claims that run counter to GR itself yet have made zero effort to show mathematically where GR is wrong using mathematics and not mere assertions. However as you stated your not proficient in the mathematics of GR so quite frankly stating others who are proficient in the mathematics is incorrect isn't a very useful tactic. A good example is stating the equation Markus posted was wrong when the truth is you didn't understand the equation. Mathematics aside do yourself a favor take a beach ball and place a ruler on the surface of a beach ball without deforming the ball The point of contact of the ruler to the ball surface is the only portion (localized region) where the Ricci curvature can be approximated as zero. That is only region where the Minkowskii metric will work. You cannot globally apply Minkowsii to a curved manifold. Precisely as Markus has been stating. PS the point of contact (tangent to the surface) is where the affine connections reside and you are using covectors /contravectors under GR. Now think of your ruler as the basis tangent vector for your lightcones give you a hint why the lightcones are changing angles ? Now move that tangent vector around the circumference of the ball ( that's what is being described by the Levi-Civita affine connection) as well the geodesic equation in curved spacetime. The kronecker delta only applies for the Euclidean portions (localized hyperslices ) with regards to Minkowsii (The RULER is the particles momentum) the surface of the ball is the Worldline.

Surveys is how you get a mean average response study on a populace of ppl. It's commonly done under phycologist studys

Surveys for starters Then where is the physics ? The entirety of physics requires at some point measurements and mathematical relations hence requiring units Here is the nitty gritty detail. Every physics field you mentioned above has momentum terms they can all be described for all its interactions via kinematics. Specifically under the Euler- Langrangian. How are planning to integrate emotion to kinematic action ?

Ok apparently a different approach is needed. @externo I would like you to consider the following we all agree that particles follow geodesics just as we all agree that the massive particles follow a different geodesic than a massless particle. So here is what you need to figure out What are the affine parameters of the null geodesic with regards to the Schwarzschild metric and PG coordinates and why are affine parameters needed in the massless particle case as they are not needed in the massive particle case. You can find the answers to the above in pretty much any GR textbook though I recommend Wald, Mathius Blau or Sean Carrol recall in the null geodesic case \[ds^2=0\] here is Sean Carrols lecture notes https://arxiv.org/abs/gr-qc/9712019# here is Mathius Blau's http://www.blau.itp.unibe.ch/newlecturesGR.pdf I would also highly suggest you look at the Levi-Civita affine connections and the Kronecker Delta with regards to Riemannian manifold with the above

Your welcome. If I may make an assumption your reason for asking is solar panel efficiency. So the article should prove useful in that regard.

Yes there is a possibility of the Talbot effect with regards to nanostructure solar panels. However with the details you have provided there isn't sufficient information to even run calculations in regards to your other question. So the best I can do for you is to link you a case study article. https://onlinelibrary.wiley.com/doi/full/10.1002/pip.3463 in essence it will depend on materials used, article mentions a couple of different options, layering thickness etc

Reading over this thread I really have come to the conclusion that although you may or may not understand SR via Minkowskii (remains to be seen) its become evident that you really do not understand GR and curved spacetimes regardless if its the Schwarzschild metric or PG coordinates. Markus has had to repeat the same statements over and over again in particularly that there is no global simultaneity. This is the primary reason the infalling clock (proper-time follows the null geodesic worldline of the metric.) that is true in both the Schwarzschild metric and the PG coordinates. You already disagreed with the mathematics I posted earlier showing that the in-fall signals is symmetric with the outgoing signals simply because you feel its wrong but have yet to actually post any mathematics showing GRs treatments to be incorrect. What you termed the conventionalist view. The other reason I feel you do not understand GR is the following statement I'm seriously hoping the nature of this question is due to English not being your primary language but anyone who knows relativity should be able to answer that question. As Markus has repeatedly stated The Schwarzschild metric is not Minkowskii.

Sounds very similar to numerous back reaction examinations without a direct reference its difficult to know for sure but sound familiar. However I can't recall the articles I may have come across it.

Do you honestly mean that you do not see the 4 coordinates in the equation for the reference frame supplied by Markus ? Specifically this term \[x^\mu\]

now that is the right attitude +1

great so you have given up on self improvement in becoming skilled in your own framework because that is what that statement means to me

My questions still remains Why rely on some program instead of taking the effort yourself? What do you gain from it ? How does it improve your skill set on the topic ?

what benefit ? nothing you have here is of any level to be beneficial. To get others to listen to any theory you need to be able to show you, yourself understand a given theory and have sufficient references, and that you understand your theory at a level where others can come directly to you and not some program for answers otherwise what is the point ?

Self adjoint ODEs \[\acute{p_0}(x)=p_1(x)\] \[\mathcal{L}=\frac{d}{dx}[p_o{x}\frac{d}{dx}]+p_2(x)\] \[\mathcal{L}u=\acute{(p_o \acute{u})}+p_2u\] integration by parts \[\int^b_a=v^\ast(x)\mathcal{L}u(x)dx=\int^b_a[v^\ast\acute{(p_o\acute{u})}+v^\ast p_2 u]dx\] \[=v\ast p_o\acute{u}]^b_a+\int^b_a[-(v^\ast)p_o\acute{u}+v^\ast p_2 u]dx\] second integration to be continued. self reminder of goal Dirichlet and Neumann boundary conditions applicability to Chebyshev differential equation first and second order polynomials. (via Sturm-Luiville)

that amounts to a handwave for I don't want to take the effort to perform my own research and seek the answers myself. How does that help you ? You can search every post I have ever made on this forum. Not once will you find me starting a thread asking a question in regards to physics. At most I ask if anyone has come across a good article on a topic. I know how to perform my own research and know how to find the answers to any question I may have for any particular research interest I have ever had. I never rely on others for any study.

You would have a very hard time convincing me of that. For that matter we also have very rules involving ChatGPT on this forum. However that is irrelevant The person that should become the expert in your theory is you. Yourself not some computer generated response. How do you plan to do that if you rely on others including ChatGPT

quite frankly ChatGP will lead you down so many garden paths its next to useless. If you want to rely on that. That is your choice. I for one rely on hard work and actual studies than some computer generated response. Its the only true way to understand any topic. A good example of the number of tradespersons I have met that have such a reliance on calculators they don't even know how to add fractions or divide fractions. I've lost count on the number of times I've run across that

They are examples you asked where mathematics can be applied. another method being correlation functions involving statistical mean averages on other case studies. It is quite possible to employ professional grade examinations on any study if you understand the techniques even philosophy.