Relativity 110e: Cosmology - Perfect Fluids, Cosmic Rest Frame, Equation of State

[@kimchi_taco]

If all science lectures are like eigenchris video, human civilization must be already multiplanet.

[@Schraiber]

Fantastic work as always. I think I need some time to understand the cosmic rest frame, and I really appreciated how you contrasted it with the aether

[@fisicadubao]

Congratulations. Excellent class!

[@thenimbo2]

It must be noted that the cosmological principle is a postulate. There are observational data challenging it on large scales. Isotropy and homogeneity are not static, so the universe being such at T=0 doesn't imply it'll be so at T=now. It does vastly simplify the equations however.

[@robertengland8769]

This kind of video makes all this a little more understandable. Thank you.

[@spongedinosaurs]

these videos are awesome and fun... seriously

[@punitsolanki5744]

Excellent work ........💯💯😍

[@klevisimeri607]

Amazing!

[@Opt1987]

So simple assumptions of the energy momentum tensor give rise to governing field equations which can be used to describe radiation/matter movement through large distances of space. The cosmic rest frame gives clarity on the frame the matter is moving relative to makes sense.

[@vazp3]

thanks, the video helped me a ton :)

[@TheMemesofDestruction]

Thank you so much for this! ^.^

[@punitsolanki5744]

Keep it up bro💯💯

[@jasonrejman1956]

Thank you for all of your hard work in these endeavors. I am left wondering whether we are guaranteed a rest frame for the CMB in which no Doppler shift exists. Could you kindly indicate any evidence of this very interesting frame? Here's where I'm stuck: given a 3-sphere embedded in R4, all points on the 3-sphere have the necessary homogeneity and isotropy of a perfect fluid. I'm trying to convince myself, WLOG, that these characteristics are inherent (or at least can be interpolated from current data) to the universe writ large. A measurable Doppler shift with respect to any point on said 3-sphere could be translated to a shift at any other point, with one of two outcomes. (1) the Doppler shift is zero everywhere. Or (2) the Doppler shift is non-zero everywhere. I find an empty Kernel in every attempt. Any guidance is very welcome.

[@FullAfterburner]

37(n+n+n)=nnn for 0 < n < 10. Sin (666) + Cos (6*6*6) = -1.618.... (golden ratio electromagnetic helix). P=NP. I solved all seven. It was marginally difficult. Takes a good while to explain.

[@FullAfterburner]

The Black Hole Time Traveler. God bless. Millennium Prize Problems. My short explanation. Not long. P=NP is 37(n+n+n)=nnn (for n=1 to 10). 12th Prime is 37 for Fine Structure Constant 137. 137.5 degrees counterclockwise to ~37.7 at the 37th minute of the hour. 13th Prime is 41 for fine structure constant 139. 216 deg. Gabriel's Horn Mathematics. Sin (666) + Cos (6*6*6) = -1.618... (Golden Ratio Riemann Zeta Function Electromagnetic Helix, Dirichlet's Theorem Prime Spiral). Euler's Identity & Mandelbrot Set Equation & The Irrational Numbers To P=NP. New Trinity/Triplex Number System from 1 to 10 and so forth. (-80538738812075974)^3 + 80435758145817515^3+. 12602123297335631^3 = 42 Ergosphere of a Black Hole. 1/2 turn Mobius Strip = 1/2 Critical Strip Fermions and electrons 1/2 spin. www.twitch.tv/blazer728 docs.google.com/document/d/e/2PACX-1vTUxpfr7MrUU8wIn1CBNvbJRM3CA4OLk3Co_5JdTCBIAyECvt2CxJEVBPBUIZbU2qTWZ8iaqiXkDcrE/pub - Human Conscious Book gnosticwarrior.com/serpent-on-the-cross.html

[@FullAfterburner]

kzbin.info/www/bejne/ql6yXoSanZyJrLM&ab_channel=minutephysics - The Unreasonable Efficiency of Black Holes (42%)

[@FullAfterburner]

One word: antiprotons en.wikipedia.org/wiki/Penning%E2%80%93Malmberg_trap

[@steamhunter7504]

Hi! Thank you for making videos about Tensors and Relativity, they are very helpful! I have one question: How much of your Relativity course is left? How many more video lessons are expected?

[@eigenchris]

A few videos are missing, but the series is basically over. The ones I have left to make are: 106b - summary of Tensors (already discussed in my Tensors for Beginners videos) 108e - Gravitational Frequency Shift 110d - FLRW Geodesics for expanding universe 110f - Friedmann Equations Derivation

[@steamhunter7504]

@@eigenchris Thank you for your quick response! Do you have any plans, other subjects, after Relativity?

[@DmAlmazov]

Will you make videos about principle of least action and Lagrangian mechanics? All textbooks are using it to derive Einstein's field equation (and mostly all equations). It seems more physical than your derivation from contracted Bianchi's identity, which is mathematically correct, but physically it is not.

[@eigenchris]

I probably won't. It may be due to my lack of experience with Lagrangian mechanics, but I find equations derived from action principles to be hard to wrap my mind around. I tried to present the EFE as a natural extension of Poisson's Equation once you apply the rules of special relativity to it. The derivation is a bit cumbersome, but it builds on previous concepts. Conversely, I have no idea where the Einstein-Hilbert action comes from, and while it's nice to confirm the EFE come from an action principle, I don't find the derivation very instructive.

[@zwonderfulz]

Sir, honestly, your explanations is amazing. I am really appreciated for your works. Could you consider differential geometry on the aspect of Lie group or AdS/CFT for the next series?

[@eigenchris]

I don't know anything about the AdS/CFT correspondence, so unfortunately I can't. I was planning on making a series on spinors, which would involve Lie groups to some extent.

[@zwonderfulz]

@@eigenchris Can't wait to see the contents on spinor. This is also very much exciting. Always support you sir. Thank you very much for creating these fantastic videos.

[@Rihanna8K]

@@eigenchris A series on spinors would be amazing!

[@eigenchris]

@@Rihanna8K It's going to be based on this video (and the 6-7 hours of talks I gave, linked in the description). However, these talks are disorganized, and even have mistakes. So I'm going to remake them into a proper series. But you can watch these if you want a sneak peek and you find them tolerable. kzbin.info/www/bejne/hWKVimN6oLF3nJo

[@Rihanna8K]

@@eigenchris Thank you for the answer! I love your work! I'm an undergrad physics student and I'm currently taking your tensor calculus course since my university does not offer a proper one 😂

[@mariafernandalarrotta7805]

Thank you very much for this class and the whole playlist. I cannot understand why the energy-momentum tensor is defined as a contravariant tensor. Could you please explain it to me?

[@eigenchris]

There are multiple versions of the EM tensor, with upper indices and lower indices. The version of upper indices can justified by viewing the one upper index as being related to 4-momentum (which has an upper index) and the other upper index as being related to the number-flux 4-vector (which has an upper index, and tracks the density of particles). So you need to upper indices to keep track of both momentum and particle density. I cover this in my 107e video, if you want a review.

[@bahmann592]

Excellent video! Question-We have two energy sources & each warp spacetime & has its corresponding metric tensors. To find the metric tensors at specific locations in spacetime, when the two energy sources are combined together--do we just add the metric sensors corresponding to each energy source?

[@eigenchris]

Generally speaking, you can't do that. The idea of adding two solutions to get a third solution is called "linearity". Schrodinger's equation from quantum mechanics, and Maxwell's Equations for electric and magnetic fields, are examples of linear equations where you can add solutions like this. The Einstein field equations are non-linear so generally speaking you can't add solutions.

[@ruedigersens9888]

@@eigenchris I'm an old German quantum chemist who is gradually fighting with the math I could do more easily when I was young. I would like to encourage you to explain the scenario of Inflation before the surface of last scattering happens to get past light cones of different observers to overlap such as to solve the horizon problem. Sorry for just asking you in this direct way; here is an article I'm just struggling through: The Shared Causal Pasts and Futures of Cosmological Events May 2013Physical Review D 88(4):044038 May 201388(4):044038 DOI:10.1103/PhysRevD.88.044038 SourcearXiv You're great.God bless you!

[@tommywhite3545]

Great.

[@bahmann592]

Great videos as always. I am a great fan. For a fee, do you provide services to develop a mathematical/scientific analysis & evaluation of a specific cosmological model?

[@eigenchris]

Thanks. I don't think I'm an authority on that. My understanding of GR doesn't go much beyond a standard 1st year graduate course.

[@asrafali8093]

Hello Chris, What if we solve geodesic equation by FLRW matric, I mean what we will get? Description of expansion of universe?

[@eigenchris]

That is what the 110d is supposed to show. The most basic timelike geodesics all have constant position for all time. So the FLRW coordinate basically "expand along with the universe". But there are some follow-up questions I haven't found the answers to yet, which is why I skipped that video.

[@aninditagupta7370]

Hello sir, I've watched your videos. If you can make a video on Gravitational waves where the energy momentum tensor is non zero and we need to use Green's function to solve that, I'll be grateful to you. Thank you.

[@eigenchris]

Thanks. Unfortunately I don't know how to do that. I've only done the basics of gravitational waves in the 109 videos. I've been making relativity videos for 2.5 years now and I plan to stop and take a break after this month.

[@Handelsbilanzdefizit]

So, choosing "Cosmic Rest Frame" is just for convenience? Because we use tensorequations that can be transformed, we could use every frame for this "PerfectFluid-Cosmology"

[@-_Nuke_-]

Same question, no answer yet...

Great Video. Where is the 110d?

[@eigenchris]

I made a community post a couple weeks ago explaining I'm struggling with the concept of gravitational redshift and cosmological redshift, so I'm taking a little extra time to ensure I have those concepts right, and posting other videos in the meantime.

[@neopalm2050]

It's a bit of a surprise that the rest frame of the CMB would also be the zero-momentum frame near every point. I get that at the time the CMB happened, the universe was incredibly uniform. I suppose that total uniformity is both a high entropy state and a state that's preserved by GR, but the consequence that around every point in the modern universe the zero-momentum frame is the same is a bit of a shock.

[@eigenchris]

It's only when you average over very large scales. The models presented in these videos only work reasonably at huge scales. At the scales of individual galaxies or star systems, our assumptions about homogeneity or isotropy fail.

[@-_Nuke_-]

@@eigenchris hmmm that should have been clarified except if I missed it

[@osamahasan3288]

hi, Mr.Chris, would you kindly direct me to to relativity lecture 110d.It seems missing > Thanks for your well presented and clarifying lectures.

[@eigenchris]

It doesn't exist yet. I mentioned in a community post a couple weeks ago that I'll be uploading the remaining videos out-of-order since I'm still working on understanding some concepts.

[@osamahasan3288]

@@eigenchris thanks for replying. i heard from you when you downloaded spinners but i didnt fully understand since i thought things related to quantum mechanics only. Best of luck for all.

[@bahmann592]

Is there an EM tensor for vacuum energy due to quantum fluctuations?

[@eigenchris]

I don't know QFT very well so I can't answer.

[@bahmann592]

@@eigenchris Thx! In Min 22:04, you mention that incase of w=-1, EM tesnor is (density c2)---why is this considered a constant, as density varies and decreases with time as scale factor increases. In other words can we have a diluting fluid(density decreases with time) that has w=-1

[@eigenchris]

@@bahmann592 In the next video on the Friedmann Equations (110f) I'll go through a derivation showing that for w=-1, the energy does not dilute as the universe expands.

[@physicsisnice4866]

I love you

[@steffenleo5997]

Good Day Chris, could you please explain result these tensor product of U_u * g^uv*U_v ; where U_u and U_v are four velocity and g^uv is contravariant metric tensor. I read in other literature(Physics book) result is - 1? Have a nice weekend... 👍

[@eigenchris]

So, I use the (+ - - -) metric signature for the metric, so timelike vectors (such as the 4-velocity vector) will have positive squared length. If you use the (- + + +) metric signature, it will have a negative squared length instead, so this is why your result is negative. Also, some textbooks use units which set the value of "c" to 1. So instead of saying c = 300,000,000 m/s, they'll use c = 1 light-second/second. The 4-velocity normally has a length of "c", but if they use the convention where c=1, then the length with just be 1, (or -1 for the - +++ metric convention).

[@asrafali8093]

Eigenchris, it would be good if you derived the EM tensor for perfect fluid. Will you?

[@eigenchris]

For the SR case at the beginning, there isn't much to derive. You just declare a fluid at rest with uniform pressure and no viscosity. Thee diagonal EM tensor is the result.

[@asrafali8093]

@@eigenchris Understood, but I still have a question that, how can a perfect fluid exert pressure? Because perfect fluid's particles has zero velocity, so there is no chance to exert force to each other. And in the case of cosmology, there are no two salestial body exert a forces to each other.

[@eigenchris]

@@asrafali8093 The pressure is due to radiation (and to some extent individual particles traveling close t the speed of light relative to the cosmic rest frame). The CMB radiation is traveling in all directions at all times, so there is momentum traveling in all directions overtime. This is the source of the force/pressure.

[@KaliFissure]

Why is the cosmological principle not expressed or consistent temporally? Was this why Einstein believed in steady state?

[@eigenchris]

At this point we have evidence that the universe is changing in scale overtime (the redshift of distance galaxies), so we only assume the universe is the same over space, not overtime.

[@KaliFissure]

@@eigenchris do we though? BranDicke metric evolved a Doppler effect the same as Hubble but simply from the curve of spacetime because we exist in local singularity called observable universe. Plug mass of universe into Schwartzchild radius calculator and you get 13.8 billion years in radius. Not age. The age was tracked on.

[@souvikmandal1989]

I can't find the relativity 110d video. Will it come later?

[@eigenchris]

Hoping by the end of July, give or take. I just finished 110f, so 110d will be my last relativity video.

[@eugenioguarino2651]

Well Chris, I can't help making the following simple deduction: if a point (and a CRF with it) exists in the universe where the average velocity of everything is zero, this must be the same point where the Big Bang took place. As we accept postulates of homogeneity and isotropy, that point cannot have moved since the BB, while matter, light and space went on expanding the same way in any direction. It seems a simple logic consequence of the postulates. I know that this looks like a philosophy issue rather that a physical one, but it also seems logically untouchable. What's your thought about?

[@eigenchris]

If you "turn the clock back" for all geodesics in the universe, they can all eventually be traced back to a singularity at the big bang, where the scale factor is a=0. We don't full understand what this singularity means right now, and it might not be physical. But this is what the math of the standard version of general relativity tells us.

[@scp_researcher953]

thanks sir im about to wtite my scientific paper where is the last video

[@eigenchris]

Are you talking about 110d (the one on FLRW geodesics, and cosmological redshift)? It will be out in a week or two.

[@scp_researcher953]

@@eigenchris let me give you an idea perherps Einstein forgot to think of, the rate of progress of time in a frame with reference to the observers reference frame, this could tell us about the speed with which time moves forward in any reality, try to think about it, its just my genuine thought 💭

[@scp_researcher953]

can you please do QUANTUM MECHANICS

[@eigenchris]

I don't understand QM enough to do a series on it. Sorry.

[@scp_researcher953]

@@eigenchris Apologies your honor, I had no idea, I'm sorry

[@chritophergaafele8922]

Where is video 110d

[@eigenchris]

I haven't made it yet. I'm making 110f now. 110d will come later.

[@Sergiypsm]

Why next video (43d) is private?

[@eigenchris]

@@Sergiypsm 110f is currently private because I'm still reviewing it for errors. I'll release it soon.

[@amonraa7329]

Where is 110d?

[@eigenchris]

I haven't made it yet. I made a community post a couple weeks ago explaining that the last few uploads will be out-of-order, as I'm still trying to figure some of the physics concepts out.

[@-_Nuke_-]

16:00 So since the cosmic rest frame applies to everything - we can always tell if someone is stationary in space or moving in space with constant velocity or accelerating relative to the CMB? EDIT: I guess that's wrong, since this works only in huge scales and not in the scales of galaxies and stars. Which renders the name "cosmic rest frame" frankly and pedagogically - terrible.

[@eigenchris]

We can measure the velocity of specific galaxies relative to the CMB. For example, the Andromeda galaxy is actually moving towards our own Milky Way galaxy, so it's not moving perfectly along with the CMB coordinates.

[@-_Nuke_-]

@@eigenchris So can we use this to make our life easier? I mean the twin's paradox becomes trivial if we compare the traveling's twin motion against the CMB. etc... Can we use that cosmic reference frame to create a spacetime clock, that will always measure our motion and relativistic effects of our spaceship against that CMB?

[@eigenchris]

The existence of the CMB doesn't change any of the laws of relativity. It's just a convenient coordinate system to use to measure the expansion of the universe. The twin paradox works the same way, regardless of whether or not the CMB exists.

[@-_Nuke_-]

@@eigenchris Yes but its just so much easier no? We have 2 twins A and B Twin A stays on the Earth that based on the CMB, it travel at 1% C towards a point in space name it Ω... Twin B gets into a spaceship and travels with 90% C relative to the CMB towards the same Ω point. The fact that B will age less than A is obvious, since 90% >> 1% relative to the CMB. We know that the faster you move in space relative to someone else the slower you move in time relative to that same observer; No need to invoke changing frames, or acceleration, or gravity, or rotating plains of simultaneity to explain it at all... Twin B travels at a greater speed in space relative to the CMB, than Twin A does relative to the CMB, therefore it ages less. The twin doesn't even need to return, for their clocks to be compared for us to know that. And most importandly Twin B can't claim that it itself is stationary and that the Earth is moving away from him at 90% C as he can in the normal twin paradox... To me this seems like CMB has made our Universe not relative. Sure you will still experience relativistic effects, but these are more like optical illusions now. What am I missing? Cuz I like this version of reality so much better!

[@NoNameAtAll2]

I feel like the video got too much into making CMB the athority CMB is just the representative of all matter/energy and it's the "average energy" that is defining rest frame

[@NoNameAtAll2]

maybe not energy, but momentum but I hope my point is understandable...

[@eigenchris]

@@NoNameAtAll2 I felt I needed to justify its existence because the energy-momentum tensor and a(t) factor used to get the Friedmann Equations don't make sense unless they are relative to a specific frame.

[@NoNameAtAll2]

@@eigenchris yeah, and I'm saying that that frame comes from average universe state first and we're detecting it through CMB second

[@cykkm]

@@NoNameAtAll2 The main point is to define the special frame, and much less so how, or even whether or not we detect it. Essentially, the choice loops back at the Cosmological Principle symmetries underlying the FLRW metric, so you want the a(t) to parametrize evolution of the whole universe in the "most symmetric" frame in the same sense. Being in fact detectable, it also has a well-defined physical meaning, and you are correct that this comes second. There is nothing wrong with that order.

[@haniamritdas4725]

So essentially the Cosmic Rest Frame concept clarifies the 4D context of spacetime. This is a definition of the aether, not a contradiction of the concept of a medium of space. I think these ancient philosophical battles are rather tiresome hairsplitting up to the point of applying mathematics to the data. Then whether you call the medium an aether or not is just word choice, not about the substance of the theory.

[@-_Nuke_-]

Light doesn't need a medium to travel into, since light exists at its destination. (Light experiences no time and no lenth - for light spacetime is 2d) So no medium required.

[@haniamritdas4725]

@@-_Nuke_- That sounds like memorization to me. I have heard all the same things, but I observe that light still takes time to travel, so light is traveling through time as a medium, and space as a component of time. These two are both forms of substantial energy and in combination with matter bend the trajectory of light by changing its velocity in the same way as passing through a dense medium like air or water or glass changes its velocity. The idea that light travels at a constant velocity in "free space" was a good enough idea as long people had an empty-minded view of spacetime; which is never "empty" itself since it _is energy_ . Light expands into 4d spacetime, which is a quadratic pair of quadratic planes, not just a 2d manifold. These neat little statements tie up the universe in a box, don't they? But do you understand what you said any better than you do what I have said in response? Or have you just heard it a million times? I don't think we have the answers down so pat yet.

[@KaiseruSoze]

An empty universe is a singularity. The three examples you give are pure abstractions. There is nothing in them. No rulers and no clocks.

[@eigenchris]

You can still measures proper lengths and proper times along geodesics in an empty universe. As our universe expands, its density will approach zero, and the "de Sitter Space" I mentioned becomes a good approximation of the universe.

[@cykkm]

“An empty universe is a singularity. The three examples you give are pure abstractions.” - This inadvertently implies that an empty universe is _not_ an abstraction, but rather something physical. Ours isn't (of matter-energy, I assume?). Also, the singularity has a precise technical definition in the scope of PDE math; and you're correct, one can construct a hyperbolic space where solutions to EFE with a \Lambda term and zero stress-energy are singular everywhere-but that's also a pure abstraction. It's much clearer to speak about an empty spacetime than an empty universe (unless arguing Mach, which I'll gladly pass, please). It's empty but not scale-free because _c,_ thus metrizable, save for curiously exotic topologies. As for the rods and clocks, BYO. :)

[@eugenioguarino2651]

I understand that this would mean that the universe has a center and that's an unacceptable idea, but...