Mordred

Take a close look at the Lorentz transforms in terms of distance with time as an interval (ct) as you increase velocity you have an inverse relation with proper time also you get length contraction. Consider further velocity itself requires a start and end point with a rate for the object to transverse that distance. That should help you understand why time has the opposite metric signature to space components.

Seems the OP is looking for what can be described as fundamental reality. The answer may give you nightmares but let's do a step by step process of elimination. Matter which are fermions are not little bullets (not corpuscular) meaning they are not solid but are described as field excitations. So one may naively believe fields are fundamental. However this isn't true either. All fields regardless if its spacetime or any other field is an abstract descriptive of a distribution ( a collection of values measured or mathematical) So one then goes onto energy or mass being fundamental ( well again this wouldn't work ) energy and mass are properties of a state or system being described. They do not exist on their own. So what are we left with ? We'll your really left with little more than configurations. Every previous descriptive while measurable ( particles, fields, energy and mass) are fundamentally convenient descriptives and there are literally hundreds if not thousands of papers arguing the above cases in metaphysics arguments etc. In so far as how does one describe configuration space ? Well the closest I can see is quantum information but this once again becomes abstract as the methodology treats information as on/off switches. Given the above from my viewpoint " What is fundamental :" is a question that is currently unanswerable and may likely always remain unanswerable. We can only measure so deep regardless of how precise our equipment gets. ( cannot measure below Planch length) . In QFT for example the observable measurements require a minimal 1 quanta of action. This is where the term virtual particles arose which is really a bookkeeping device for propogator action ( the internal wavy lines of Feymann diagrams). Hope I wasn't too far off base on the assumption the OP was looking for what is fundamental reality. Its rather tricky to tell ( you may notice I used the term configurations ) this term does not label what the configuration is as any label is also abstract. If this helps physics describes what we can measure. It speculates on what we cannot measure based on what we have measured. In short given the above there is no fixed entity. The best we can hope for is accurate descriptions of relations between abstract objects such as a state. (set of configurations)

I was browsing and chanced upon a very straightforward and well described video, that is rather infectious in the sheer enthusiasm of the Presenter. It covers tensor ranking and the requirement for tensor usage using conductors. Thought I would share it as it was a very enjoyable and informative video to watch in very simple treatments that is easily understood.

In terms of mathematical proofs the following properties should be followed Concise (not unnecessarily long) Clear (not ambiguous) Complete (no missing intermediate steps) Logical (every statement logically follows) Rigorous (uses mathematical expressions) Convincing (does not raise questions) The way a proof is presented might be different from the way the proof is discovered. Proofs will include defining the limits of a set, step by step derivatives to answer questions such as -How did you arrive at this equation ? - What limits does the equation have ? -does the equation follow other related and well established equations that do the same operation. If I was developing a new theory, methodology or set of equations the above is the steps I would follow.

your welcome

Well I don't really go into the various struggles Einstein had nor what his thoughts were and much of the information one can find on the internet is often misleading. However I have seen numerous papers suggesting he struggled with Minkowskii 4d treatment in that he preferred to keep space and time distinct. I also saw various papers involving Mach principle which suggests that space should not exist independently but has a dependency on matter distribution. There are others relating to Block universe etc. example here https://www.sciencedirect.com/science/article/abs/pii/0039368194900566 for the Mach case

I will have to wait till Monday to confirm but it looks like the Horndeskii relations were being employed to pull time elapsed data from the spectrographic filtration software and have nothing to do with the software calibration itself.

Unfortunately I would have to agree with you.

@MJ kihara Type the @ symbol then start typing the name it will pull a drop down elimination list. I did the above from my phone

Sounds like we may be on the same page or similar enough that distinctions will become apparent in the discussion. SR and GR uses the interval (ct) to give time dimensionality of length. However length can have several meanings in SR and GR example conformal length and proper length so better clarity which specific length is being examined will be needed. As far as curvature terms I would consider the amount of curvature as emergent as the stress energy momentum tensor that tells spacetime how to curve contains emergent properties such as the pressure term. Subsequently the ds^2 separation distance between observer and emitter would also be emergent

Well in physics an emergent property is a characteristic of any complex system. They arise from the collective interactions of the systems components and are not additive. There are some theories that ask the question is spacetime emergent if so then what properties . A decent listing can be found here https://arxiv.org/pdf/0711.4416 so given the above how would you determine the collective interactions leading to the meter ?

The opening post wouldn't be useful to answer the question "if geometry is fixed or emergent. Under SR and GR no coordinate system is privileged one can arbitrarily change to any coordinate system and the physics remain the same. This is where one can apply for common example Pythagorous theorem as any curved spacetime the angles will not equal 180 degrees. Positive curvature the angles of an equilateral triangle will be greater than 180 degrees while negative curvature it will be less than 180 degrees. However in each case there is a method to define a small region where it is spatially flat. It is likely this consideration that Einstein worked with others such as Minkowskii to help resolve. Emergent properties has rather specific requirements so clarity on what you understand as emergent will be needed as a visual aid to help understand the above see http://cosmology101.wikidot.com/geometry-flrw-metric/

Thank you that is all I've been trying to do is provide challenges and other considerations not to compete with you but to provide methodological comparisons. I will assume you can use the above to determine mirror images and determine the angular diameter distance of the original object. The alternative method I wanted to see how you handle was flux vectors which also involved specific and mean intensity. example Einstein ring or different distortions as per the images in article below https://web.pa.msu.edu/people/abdo/GravitationalLensing.pdf

so far I have determined that it ties into the monopole and quadrupole moments via Legendre Polynomials \[P_I(k)\frac{2l+1}{2}\int_{-1}^1\mathcal{L}_I{\mu}P(k,\mu)d\mu\] \[P_o(k)(1+2/3\beta(bl)^2+1/5(bl^2\beta^2)P_m(k)\] \[P_2(k)=(4/3(bl)^2\beta+4/7(bI)^2\beta^2)P_m(k)\]

I am looking for decent books or articles covering Lovelock and Horndeski second order extensions to the Einstein field equations as I have a need to understand how and why its employed in line intensity mappings for a calibration project I am on. So far I've isolated few of the terms but have encountered others I still need to isolate for its functionality. I am currently studying the following but could use other recommended literature https://www.ugr.es/~bjanssen/text/Tesis-JoseAlbertoOrejuela.pdf What I'm trying to resolve is how it fits into the line power spectrum \[P_{Cluster}(k,z)=b^2(z)I^2(z)P_m(k,z)\] where I is the mean intensity. shot noise given by \[P_{obs}(k,z)=P_{clust}(k,z)+p_{shot}(z)+P_N\] with Fourier mode at scale K \[N_m(k)=\frac{k^2\Delta kV_s}{4\pi^2}\] and variance \[\sigma^2(k,z)=\frac{P^2_{obs}(k,z)}{n_m k}\] with anistropic matter power spectrum linear plane parallel approximation \[P_{obs}(k,\mu,z)=[b,(z)^2 I(z)^2+f(z)^2I(z)^2\mu^2]^2P_m(k)\] the above being a more generalized format than the one being employed but its far easier to relate to.

Wow are you ever close minded. Take a look a look at geometry simply because you replaced geometry with your space equals energy does not mean you didn't have to account for geometry. Same goes for mass using the energy momentum relation Go right on ahead keep ignoring me I really couldnt care less. Does not stop other readers from reading my posts. Those are specific examples.

Piece of evidence to consider for Hubble constant and expansion, Did you notice that Hubble constant is decreasing in value while expansion is accelerating ?

Is it onological baggage when you consider the well established detail that no massive object can have a velocity of v=c ? Yet you conclude there is no massless photons ? Thats one of the common mistakes we often see here is ppl drawing conclusions based on specific relations whether its their own or not and not looking at other pieces of evidence. Prime example being the twin paradox. It was never a paradox but was directly related to loss of symmetry between the Lorentz transformation which you are applying albeit through your beta function. Spacetime curvature your running the same risk. You do get a different result in how you treat time than Newtonian with non variable time. So ask yourself " why is spacetime curvature needed " Then look at the axiom " the laws of physics must be the same regardless of reference frame" This includes Pythagorous theorem which is something you haven't been examining ( it is a geometric relation). See the danger of not taking into consideration other pieces of evidence including those outside your own work ? Lets try a professional level of examination example. Encyclopedia Inflationaris https://arxiv.org/abs/1303.3787. Every single model in the above paper ( over 70) are still good fits to observational evidence. The order its currently in is best fit onwards. Yet none of the other fits are invalid. To narrow down the extensive list uses pieces of evidence ( NOT covered within their own models) that leads to invalidation. I'm positive you do not want others throwing out your hard work because you missed pieces of evidence that may run counter to your conclusions.

Would help to seperate invariant mass ( coupling strength) (old term rest mass) from variant mass ( relativistic mass) affected by observer Lol ever wonder where the 19 parameters come from for the standard model. Covering all the invariant mass couplings would definitely be outside the scope of this thread.

Careful on this statement invariant mass is directly related to how strongly a particle couples to its respective fields it interacts with. To be honest I wish I had time to go through your math. I simply don't, hence been using generalizations. That stupid contract where I am working on filtration software pertinent to a change in grating for the spectographic equipment of that telescope I previously mentioned has been a major headache. Turns out the software employs Horndenski gravity which its been a while since I even studied that variation. Needless to say problematic and head is mush at the end of the world day. Glad to see KJW spotted the error well done

Nice post follows some of my concerns but far better worded +1. One I've hinted at numerous times is how does one apply field treatment distributions which also happens to be one of my favorite studies in different formal methodologies. Another being how would it handle multiple unknowns if say you had to determine all the Cosmological parameters given just volume and change in volume over time. Nothing else.

Future challenge " what is the magnification" How would you use the quoted relations to determine this. As an assist to help you establish the above. https://www.google.com/url?sa=i&source=web&rct=j&url=https://kamion.pha.jhu.edu/Ay127/week7.pdf&ved=2ahUKEwiu-evdloqTAxVGFzQIHciJJsUQ1fkOegQIDBAC&opi=89978449&cd&psig=AOvVaw2blqf2txIV8JzlX7EF6dwh&ust=1772848843742000 Note the use of angular diameter distance. Just a little side note there is a lensing methodology to determine Hubble constant. See equation 9.6 https://www.google.com/url?sa=i&source=web&rct=j&url=https://orbi.uliege.be/bitstream/2268/74098/7/Refsdal_Surdej_RepProgPhys_1994_56_117_185.pdf&ved=2ahUKEwiu-evdloqTAxVGFzQIHciJJsUQ1fkOegQIDBAU&opi=89978449&cd&psig=AOvVaw2blqf2txIV8JzlX7EF6dwh&ust=1772848843742000

I was never arguing logic to begin with. I was explaining where your articles lack detail as well as the examples used. If you look closely you would recognize most of my posts have not been theory dependent but rather that of mathematical relations that need to be presented in a more readily to relate to manner. Math operations such as curl and divergence is a huge chunk of physics relations. So is ray castings when it comes to observational cosmology or non uniform mass density distributions and their effects. These are major areas that any other theory such as GR, QFT etc deal with. Its also one of the reasons why the equations used get so complex. In every theory above scalar field and scalar treatments is the easiest case scenarios. So if you really want to show the power behind your methodology try some of the examples I've provides. Worse case scenario it will help others better relate to what you have

realized I had already asked the question I had posted, earlier so withdrawing this post

Look back at how often you shot me down for mentioning common physics treatments or would you rather I go through them and post them. That last post of yours was a good example. I specifically asked the question Do you have an equivalent to the gradient operator. Did you respond with the math ?