Mordred

If the path is not possible its not included this includes faster than c. Quantum tunneling is a possible path so is inclusive as it describes how a particles can cross a potential barrier higher than their total kinetic energy so it also has a non zero probability.

One handy trick I learned in programming is that in binary divide and multiply by 2 is a bit shift left or bit shift right. Applying this operation can help reduce stack usage though it's been ages since I last did serious programming.

At extremely low temperatures the DeBroglie wavelengths of the individual particles overlap and form a macroscopic single wavelength. Relevant formula \[\lambda_{db}=\sqrt{\frac{2\pi\hbar^2}{mK_B T}}\] As both Bose-Einstein statistics and Fermi Dirac apply the Boltzmann equation specifically Maxwell Boltzmann the mathematics will typically apply phase space. Defined by the Hamilton in the mean field approximation as a single interaction term. I have often seen this associated with the Hartree approximation to give some of the math typically involved. Should add the Compton wavelength also becomes modified into an effective Compton wavelength using the equations in the link below https://en.wikipedia.org/wiki/Gross%E2%80%93Pitaevskii_equation

One detail I would like to add with regards to energy momentum. Time translations conserve energy while space translations conserve the momentum. Combining the two gives energy momentum conservation. I honestly hope you pick up on my last post orbiting bodies aren't orbiting each other. They orbit their common barycenter. With the Sun and Mercury system that's negligible that isn't true if both mass terms are similar in value. Good example Jupiter and the Sun they orbit the common barycenter just outside the Suns radius. Just a little trivial sidenot. Ever wonder how gravity wave detectors get calibrated ? They literally place a 1 ton or greater known mass beside the beam. No orbit involved yet the beam will change in angle

How will that methodology work if you are given two stars of equal mass orbiting each ? How would you determine the common barycenter for other mass variations of 2 stellar objects.

The method to obtain a BEC by using lasers would certainly do so. So will changing the state of the atom from one previously entangled from non BEC to BEC.

Did some digging and found 3 classes of solutions all three examined in this paper. https://arxiv.org/pdf/2308.07374 With Mc Vittie being more commonly referenced (Schwartzchild FLRW metric). There are a few others but the three in this paper are seemingly examined more Anyways though there is some relevance (static vs commoving coordinate embeddings) it runs the risk of derailing the thread but thought I'd share what I found.

Solid point its food for thought when I get a chance I will likely look into it in greater detail. Thanks for your opinion on an interesting problem to examine. The Desittter and anti-Desitter spacetime route may be a decent approach to the scenario to start with.

That would make for an interesting superposition state. The propagation integral would also have some interesting processes. Would definitely bring new meaning to the Greens function... Lol me too

I wonder if my wife would want those smart pants attached to her fit bit via an app lol.

That brings to mind an interesting scenario If you go from \[R_{\mu\nu}=0\] of the Schwartzchild metric to the FLRW metric which has the time component as \[R_{00}=-\frac{3}{c^2}\frac{\ddot{a}}{a}\] With space components \[R_{ij}= (a\ddot{a}+2a^2+2c^2)\frac{g_{ij}}{a^2}\] The question of the transition distance comes to mind between the 2 scenarios. I would surmise the latter would take effect when the transition between gravitational bound spacetime regions to spacetime regions not gravitationally bound. What are your thoughts on that @KJW ? Not sure why that thought came to mind but to me it does pose an interesting question

Lmao no that wouldn't be good....

Im How many times has the question of mass come up this thread ? Would not distinguishing when mass becomes relevant be useful to answer those questions ? We cross posted

So you believe yet you haven't actually stated any argument to counter my quoted statement. However that's a typical response I often see when I describe something that can be found in any textbook on GR. Feel free to go right on ahead on what you feel is right. I know GR is a field treatment and why it is a field treatment. Boils down to that key word distribution and a word I haven't mentioned is flow... If you want to test that try tracking 20000 test particles.

No you have to think GR dont restrict yourself to the Maximally symmetric Minkowskii metric. In GR the stress energy momentum tells spacetime how to curve from the Minkowskii metric. GR describes the Field distribution of the particle ensemble not individual particle motion per se but the entire ensemble. The Minkowskii metric is what is used by SR primarily which is a specific solution of the metric tensor. \[g_{\mu\nu}=\eta_{\mu\nu}+ h_{\mu\nu}\] You likely already know the Einstein field equation so I won't waste time latexing that. In the above the \{\eta\} is the Minkowskii tensor and h is the perturbation tensor the stress tensor acts upon the permutation tensor and gives the deviations from the Minkowskii tensor to result in the new metric tensor ( can often have off diagonal terms) such as a rotating star will have off diagonal terms. However then again the Minkowskii tensor isn't always maximally symmetric take observer A rotating around observer B along the z axis \[\eta_{\mu\nu}=\begin{pmatrix}-\frac{\omega^2 r^2}{c^2}&0&\frac{\omega r^2}{c}&0\\0&1&0&0\\\frac{\omega r^2}{c}&0&r^2&0\\0&0&0&1\end{pmatrix}\] The Sagnac effect I mentioned previously here is a good stepping stone to understanding geodesics, here is a quote from the following article that applies. "From this follows that the gravitational effects are locally indistinguishable from the physical ones experienced in an accelerated frame. Such an equivalence enables us to eliminate gravity in a sufficiently small region of space-time. This is done by introducing a suitable chart which supports a locally inertial frame. One can therefore assume that freely falling, inertial frames can be defined in a sufficiently small neighborhood UP of each space-time point P, and the laws of Special Relativity, which may strictly hold only at each point P, will also be sufficiently good approximations of the true laws inside UP for all inertial observers defined therein. A typical example of such a possibility is provided by a free-falling elevator in the gravitational field of the Earth. In fact, a test body inside the falling elevator will fluctuate freely as if the elevator would be placed in empty space, in a region free from any gravitational field." The article later shows how the FLRW metric equations of state are involved in the Christoffels themself further down another relevant quote "Previously, we mentioned the Einstein’s conceptual experiment in the case of one test body inside the free-falling elevator. However, if we put two test bodies (instead of one) inside the elevator, then there is an important physical difference between the two configurations previously mentioned i.e., free-fall in a given field and real absence of field, which soon clearly emerges. Suppose, for instance, that the two bodies are initially at rest at the initial time t0. Then, for t > t0, they will keep both at rest in the absence of a real external field; on the contrary, they will start approaching each other with a relative accelerated motion if the elevator is free falling. The relative motion is unavoidable, in the second case, due to the fact that the test bodies 15 are falling along geodesic trajectories which are not parallel, but converging toward the source of the physical field. So, even if the relative velocity of the two bodies is initially vanishing, v(t0) = 0, their initial relative acceleration, a(t0), is always non-vanishing" https://amslaurea.unibo.it/id/eprint/18755/1/Raychaudhuri.pdf I invite you to study the contents in greater detail. the last should highlight the distinction between local reference frames from the global geometry So tell me I established above and you agree the mass of the test particle isn't relevent. The proper acceleration is zero and it is the field distribution of mass/energy that determines the geodesic equations of motion " the stress energy momentum tensor tells spacetime how to curve" so where am I wrong in the quoted statement that you accused me of being wrong ?

incorrect the stress energy tensor tells spacetime how to curve which affects the metric tensor That stress energy tensor includes energy density and pressure terms and directly affects the metric tensor

nice try keep trying. Why don't you start with basic classical freefall and notice that the mass term becomes irrelevant. (all objects regardless of mass fall at the same rate...) Then look at the how proper acceleration=0 in geodesic motion ( free fall no external forces) Tell me does the expression "Under GR gravity is not described as a force but the result of spacetime curvature men anything to you ? https://en.wikipedia.org/wiki/Geodesics_in_general_relativity#:~:text=For%20broader%20coverage%20of%20this,(3%2DD)%20space. In general relativity, a geodesic generalizes the notion of a "straight line" to curved spacetime. Importantly, the world line of a particle free from all external, non-gravitational forces is a particular type of geodesic. In other words, a freely moving or falling particle always moves along a geodesic. In general relativity, gravity can be regarded as not a force but a consequence of a curved spacetime geometry where the source of curvature is the stress–energy tensor (representing matter, for instance). Thus, for example, the path of a planet orbiting a star is the projection of a geodesic of the curved four-dimensional (4-D) spacetime geometry around the star onto three-dimensional (3-D) space. As you like AI https://www.google.com/search?q=are+particles+in+freefall+on+a+geodesic+path&rlz=1C1VDKB_enCA1142CA1142&oq=are+particles+in+freefall+on+a+geodesic+path&gs_lcrp=EgZjaHJvbWUyBggAEEUYOdIBCjQxOTM0ajBqMTWoAgCwAgA&sourceid=chrome&ie=UTF-8 To be honest I had hoped you would have picked onto this when I had previously mentioned Newtons shell theorem. with regards to mass energy distribution.

My wife of course doesn't thank you as I'm running out of book storage space lmao

Good luck on that its common knowledge for those that understand GR. Why else do you think Ive mentioned distribution throughout this thread ? By the way your more than welcome to apply GR to prove me wrong.

@Studiot when I saw you post this book I decided I wanted a copy for myself. Recently received it and enjoying it thus far. Still studying the material

Under SR/GR all particles that are following a geodesic are in freefall. So mass is irrelevant so is proper acceleration. Where mass/energy density distribution becomes relevant is determining the geodesic itself.

Received confirmation, turned out it was related to an old constraints experiment and shouldn't have been in the filtration software to begin with. The software team is removing it. At least now I can focus on filtering atmospheric interference and pollutants from local industries such as the Pulp mill.

Another question one can ask is if mass is a primitive concept " why do different particles or other objects require greater force to accelerate the same distance than others "?

should have added to this one could also alternately use celerity (proper velocity) which is distinct from coordinate velocity. There is no real hard and fast methodology all math methods used to describe a system or state if they can produce the same results and fully describe that system or state should never have any restrictions based on philosophy. After all why should it ? Mathematics itself is a collection of bookkeeping tools it merely describes what we observe or measure and offers means of predicting cause and effect. The mathematics itself never defines reality. After all the universe couldn't care less how we measure it nor describe it.

Thanks though truth be told I never worry nor consider the reputation votes as being useful so no worries on my end lol