Relativity 110b: Cosmology - FLRW Metric Derivation (3 possible geometries)
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- Опубликовано: 9 июл 2024
- Full relativity playlist: • Relativity by eigenchris
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Ricci Curvature of 3-sphere: www.physics.mcgill.ca/~rhb/ph...
Ricci Curvature of Hyperbolic plane: thatsmaths.files.wordpress.co...
Info on Hyperboloid Model: en.wikipedia.org/wiki/Hyperbo...
Hyperbolic Geometry video by Henry Segerman: • Illuminating hyperboli...
Alex Flournoy video on FLRW metric: • General Relativity Top...
Dietterich Labs video on FLRW metric: • Friedmann Equations Fr...
0:00 Introduction
1:10 Cosmological Principle
4:03 Flat universe and scale factor a(t)
6:52 Closed/spherical universe
12:22 Open/hyperbolic universe
17:55 Unifying the 3 metrics
22:58 Reduced-Circumference Polar Coordinates
Reposting because I noticed the audio was de-synced on the previous version so that my voice didn't always line up with the slides.
Your voice is somewhat annoying. You start with higher pitch and when you get to the end of the sentence you have deep pitch. And it’s pretty monotonous.
Listen to the 3blue1brown video . The guy is very careful in how he sounds.
An abundance of clarity, much appreciated.
I never thought I'd stand a chance understanding this theory.
Now there's hope!
This is very helpful. Two comments, a minor one and a perhaps major idea: The minor one is not to include the hypersphere radius (capital) R into a(t). To keep both radii makes it easier to keep track of what is happening systematically. The major idea ( and i am not actually sure that works) is instead of trying to “imagine” one single four dimensional ball try to picture two “normal” 3D balls instead - this should be equivalent as you can define a 3D ball in terms of two 2D circles as well. This makes it more difficult mathematically, but physically it makes much more intuitive sense: In the end the metric of a spherical universe compares the (3D) “universe ball” to the “coordinates ball” that you are projecting it on. The (non-capital) (dr) is then the difference between the centers of those two balls. And that’s why in the case of a spherical universe the dr dimension vanishes in the metric for r>>dr: two balls merge into one as long as you blow them up to a radius that is far bigger than the distance of their origins. I hope i am making sense here, would love to be able to put this into serious mathematical terminology:(
Interestingly the FLWR metric (the Friedmann part, 1922) was developed before the existence of outer galaxies was proven (Hubble, 1924)
@pyropulse lol owned
But the RINSE CYCLE! They've all forgotten THE RINSE CYCLE!!!
The absolute best video on Flrw on the internet, no doubt.
Such beautiful detail. Very rare indeed!
Very interesting and useful videos for students and teachers alike!
Our manifold.
Surface(cos(u/2)cos(v/2),cos(u/2)sin (v/2),sin(u)/2) 0>u>4π 0>v>2π.
Notice that 4π, 2 full rotations are needed to complete this manifold.
Electron half spin is an artifact of this topology.
Great teacher forever...
Amazing job!
Very good and clear derivation
here's the previous version of the video ruclips.net/video/SsIue2jHcL4/видео.html
(if anyone wants to see comments or smth like that)
I'm confused, because you said the space with a constant negative curvature is like a saddle, but the hyperboloid you drew is very different than that. It doesn't seems to have one smile and one frown.
Its weird seeing him teach me advance relativity while finding him in a vid when he faced the avengers
Thanks !
Thumbnail makes me want to me “prove the earth isn’t flat” and how other geometric configurations wouldn’t work kinematically & dynamically
Nice Job! Does the "spacetime interval" change in the 3 different closed/open/flat universes, if so is there an equation for it?
The metric gives the equation for the spacetime interval. To get the length of a vector, you use the coefficients in the metric to get the right formula.
Excellent.....
Plz send more vid ....
In the hyperbolic parameterization, there should be an imaginary component on q so when its squared it makes -q^2. If not Gxx at 17:20 would be (R^2)((2cosh^2)-1) due to sinh^2+cosh^2.
Thank you so very much! I was scouring the internet to find a fix and then I was like "Wait! I should check the comment section!" and there you go! Exactly the answer that I needed!
I've been working on this for the last couple of hours and I've come to the realization that your solution won't work but it's actually not necessary. We get a minus in front of the sinh^2 term because we use the extrinsic metric for q*q which already has - 1.
At 12:11 you said that it's very much similar as the metric is coming out to be similar in both cases, my question is why is metric 3 by 3 in both, like for 3D flat space metric i was just having two variables, theta n phi, although now I'm having three coordinates, theta, phi, chi.
Is there any metric where the inhomogeneities and anisotropy is considered? Is it possible to construct such metric form our existing knowledge of cosmology?
Good question, I think! And interesting and deeply thoughtful also! 🤔
Great job！I really appreciate your work, helped me a lot to understand everything. I have a little question: When talking about the metric tensor in 17:50, where does the radius of 3D sphere/hyperbola R go? I took r=Rsin\chi (or Rsinh\chi) and found that k is not limited to ±1, but to be ±1/R^2.
The radius (squared) is just an overall factor in front of the metric. We absorb this into the a(t) factor, so k is only ±1. The "R" term becomes part of a(t).
@@eigenchris Oh I missed hearing that. I took a second thought but I'm still confused: if we do so, doesn't that mean the r of a spherical spacetime is limited under 1 (since sin\chi is at most up to 1)? If we interpret the r as some kind of "proportional radius", why don't we just put the R in it from the first place? Besides, if we take the "north pole" of the 3d sphere as the origin (since it is isotrophic so we can freely choose our starting point), r=R\sin\chi graphically makes sense.
At 5:45 you say ‘taken as a whole, this 4x4 metric represents a curved spacetime’ assuming a non-constant a(t). I assume that the same goes for the full unified metric (at 22:50)? Such that a non-constant a(t) causes spacetime curvature.
Basically I’m hearing three different kinds of spacetime curvature here :
1 - Curvature caused by mass/energy
2 - Curvature caused by the expanding universe ( a(t) )
3 - Curvature caused by intrinsic geometry (k)
I feel okay with conceptually separating mass/energy curvature from intrinsic geometry, but I’m feeling shaky about curvature caused by expansion alone. Do you have any resources that talk about this type of curvature? I feel like I rarely hear it referred to as such.
The point I was trying to make is that, even if spatial slices are flat (k=0), the 4D spacetime as a whole is curved for non-constant a(t). So yes, you're correct.
All curvature is caused by mass/energy/momentum (the right side of the Einstein field equations). When we write the metric using a(t) and "k" (left side of the EFE), we're just demanding that the metric take a certain form, based on the assumptions of the universe being spatially homogeneous and isotropic. It is ultimately mass/energy/momentum which determine what a(t) and "k" are. My next video on the Friedmann Equations will make this more concrete and show equations for calculating a(t) and "k" from energy density and pressure. Hopefully it will be done in about a week.
The Boöts void is 325 light years wide (0,27% of the universe). That stretches over 3 of these boxes (100 light years each) and looks far from homogenus to the rest. We also do not know how many voids there are. These spots most have had already a differend pre set cause verry early in the birth of the universe, even when they where made by super stars. The energie/mass most have bin much more dence there, to make these giant stars with short lifespan. Do we know what most galaxies inside the Boöts void are made of?
As u said:
"In hyperbolic plane H², each point looks like a saddle point",
Now thinking of this, I thought probably in a normal Hyperbola, each point may look like a point of inflection (as it is single variable calculus analogue of saddle point).
I took
y² - x² = c², and found double derivative which wasn't coming out to be 0 at all points,
So it is wrong to assume so or am I missing something?
A normally hyperbola is only a saddle point at one point (in the standard equation, it's the origin). The true hyperbolic plane can't be visualized in 3D space in its entirety. You can only look at small pieces of it. If you google "hyperbolic plane crochet", you can see yarn works people have tried to make to visualize it.
The non-zero riemann curvature tensor (@5:53)
Would this curvature account for some small amount of redshift? In the photons traveling from the distant past?
Such that, it is not the fabric of space expanding that is causing redshifted light, but the fact that light must travel (essentially) out of a gravity well of curved spacetime from the past.
The curvature of the FLRW spacetime is the cause of "cosmological redshift". This is what makes far-away galaxies appear more red than expected, because the space between us is expanding. Personally, I wouldn't call this a "gravity well", because that makes me think of the gravitational redshift effect due to a spherical mass, which is different than cosmological redshift. I plan for the 110d video to cover cosmological redshift, but that video isn't finished yet.
@@eigenchris thanks for the reply! I agree that calling it a 'gravity well' doesn't conjure the correct understanding of the time element involved. Truthfully, I've never been fully comfortable with the idea that light redshifts due to the 'underlying fabric of space expanding', especially when there are much more mathematically rigorous descriptions of kinetic and gravitational redshift.
The non-zero riemann curvature tensor @5:53 looks a lot like something that would produce a 'gravitational' redshift. (it looks pretty similar to the Schwarzschild metric with an additional scale factor and different g00 and g11 ...well, not totally similar I guess)
The analogy of a gravity well also breaks down as it's not possible to combine the concepts of gravitational potential with the Friedman equations. But, to me, it *feels* fairly similar. Such that the potential exists thru time, instead of space.
Anyway. Love the videos. I'm clearly still struggling.
I started a related thread about this on reddit if your are interested
www.reddit.com/r/cosmology/comments/wdnxps/photons_shielded_from_cosmic_expansion/
Looking forward to 110d
@@DanSternofBeyer The Einstein field equations is the accurate description of spacetime. All solutions to it like FLRW and Schwarzschild come from highly idealized scenarios. But they are applicable when the situation resemble the idealization. Eg. Schwarzschild applies very well in the spacetime surrounding spherical masses, such as surrounding the earth, sun or a black hole (each of these is a better approximation, because the sun is more spherical than the earth, and black holes even more so than the sun). The FLRW metric starts out by assuming all of space is literally perfectly isotropic and homogeneous - which it clearly is not, every direction in space looks different from earth. It is >close< to isotropic and homogeneous if we zoom out far enough though, as shown in the video, but even here it's not >perfectly< isotropic and homogeneous. But at that scale, the idealization of the FLRW metric is a reasonable approximate solution to get some insight.
But of course, neither metric is even close to being a "correct" solution for all observations. But the full Einstein field equations explains just about every accurate observation we can make. This is why analogies don't work well, we make analogies for these highly idealized models that approximate the true mathematical equations - e.g. "underlying fabric of space expanding" is such an inaccurate analogy. The equations are what's accurate. And to sort of jump on your reddit question in the same theme, this is the reason for the ISW. It's an effect that describes how the FLRW idealization breaks down in that case, since the real universe is not homogeneous and isotropic. Every question such as "what causes X/what is the mechanism of stretching" can sort of be answered in an annoying and non-insightful, but highly accurate way with "it follows from the Einstein field equations". The answer you got on reddit is a more insightful way to think about why it follows from the Einstein field equations. But really, the only way to fully comprehend where all analogies come from is to become proficient in the mathematics. And even then I think I can safely say nobody fully comprehends the full force of the field equations.
@@xiupsilon876 thanks for the reply. So much of science breaks down to 'The math works and we don't know why.' ...and I guess I just have to get used to that.
We are waiting for your next Video on Friedman equations when will you post it
Hopefully by this weekend.
So I understand that ‘r’ is the reduced radial Co-ordinate but then I don’t understand how the g22 term is r^2 and g33 is (rsintheta)^2 because that’s only true for spherical Co-ordinates if r is the radial Co-ordinate
Why do you say it's only true for spherical coordinates? Spherical coordinates in flat space have a zero Riemann Curvature tensor. The FLRW metric is only spatially flat if k=0.
It seems to me that the Cosmic Rest Frame (see CMB) violates the Cosmological Principle
Why do you say that?
@@eigenchris because unless in CRF you get different redshifts of CMB in different directions
Theory of Everything solution:
[Short answer] swap from Newton to Leibniz as our fundamental blueprint of the universe.
[Long answer] I contend Gottfried Leibniz was correct about the fundamentals of our contingent universe and he just lacked 2022 verbiage and Hamilton's 4D quaternion algebra.
More importantly is that humanity chose Isaac Newton's "real" universe, calculus, gravity, etc. This was a big mistake. We need to correct this problem.
Finishing what Leibniz started (with the intention of destroying what Newton started):
[Math; Geometry 0D point]
A point is a 0-dimensional mathematical object which can be specified in -dimensional space using an n-tuple ( , , ..., ) consisting of. coordinates. In dimensions greater than or equal to two, points are sometimes considered synonymous with vectors and so points in n-dimensional space are sometimes called n-vectors.
1D = line, straight; two points; composite substance; matter
《0D (point) is exact location only; zero size; not a 'thing', not a 'part'; Monad》
Monad (from Greek μονάς monas, "singularity" in turn from μόνος monos, "alone") refers, in cosmogony, to the Supreme Being, divinity or the totality of all things.
The concept was reportedly conceived by the Pythagoreans and may refer variously to a single source acting alone, or to an indivisible origin, or to both.
The concept was later adopted by other philosophers, such as Gottfried Wilhelm Leibniz, who referred to the monad as an elementary particle.
[Quantum]
Quark is a type of elementary particle and a fundamental 'constituent' of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei.
What is another word for quark?
fundamental particle, elementary particle.
Do quarks take up space?
Its defining feature is that it lacks spatial extension; being dimensionless, it does not take up space.
How fast do quarks move?
the speed of light
[In mathematics, a tuple is a finite ordered list (sequence) of elements. An n-tuple is a sequence (or ordered list) of n elements, where n is a non-negative integer. There is only one 0-tuple, referred to as the empty tuple. An n-tuple is defined inductively using the construction of an ordered pair]
1st four dimensions are 0D, 1D, 2D, 3D ✅.
1st four dimensions are not 1D, 2D, 3D, 4D 🚫.
Human consciousness, mathematically, is identical to 4D quaternion algebra with w, x, y, z being "real/necessary" (0D, 1D, 2D, 3D) and i, j, k being "imaginary/contingent" (1D xi, 2D yj, 3D zk).
1D-9D 'contingent' universe has "conscious lifeforms" (1D xi, 2D yj, 3D zk)..."turning" 'time'.
[In mathematics, a versor is a quaternion of norm one (a unit quaternion). The word is derived from Latin versare = "to turn" with the suffix -or forming a noun from the verb (i.e. versor = "the turner"). It was introduced by William Rowan Hamilton in the context of his quaternion theory.]
[Math; 4D quaternion algebra]
A quaternion is a 4-tuple, which is a more concise representation than a rotation matrix. Its geo- metric meaning is also more obvious as the rotation axis and angle can be trivially recovered.
How do you make a quaternion?
You can create an N-by-1 quaternion array by specifying an N-by-3 array of Euler angles in radians or degrees. Use the euler syntax to create a scalar quaternion using a 1-by-3 vector of Euler angles in radians.
"Turn" to what, you might ask. 5D is the center of 1D-9D. The breadth (space-time). All 'things' and 'parts' are drawn to the center, the whole. (The Dawn -Book of Cain on the creation of the contingent universe)
[Contingent Universe]:
3 sets of 3 dimensions:
(1D-3D/4D-6D/7D-9D)
The illusory middle set (4D, 5D, 6D) is temporal. Id imagine we create this middle temporal set similar to a dimensional Venn Diagram with polarized lenses that we "turn" by being conscious.
Which requires energy. 3D height symmetry/entanglement with 9D absorption is why we are "consumers", we must consume/absorb calories, and sleep, to continue "to turn" 'time' (be alive).
1D-3D spatial set/7D-9D spectral set overlap creating the temporal illusion of 4D-6D set.
According to theoretical physicist Carlo Rovelli, time is an illusion: our naive perception of its flow doesn't correspond to physical reality. Indeed, as Rovelli argues in The Order of Time, much more is illusory, including Isaac Newton's picture of a universally ticking clock.
Does time exist without space?
Time 'is' as space 'is' - part of a reference frame in which in ordered sequence you can touch, throw and eat apples.
Time cannot exist without space and the existence of time does require energy.
Time, then, has three levels, according to Leibniz:
(i) the atemporality or eternality of God;
(ii) the continuous immanent becoming-itself of the monad as entelechy;
(iii) time as the external framework of a chronology of “nows”
The difference between (ii) and (iii) is made clear by the account of the internal principle of change.
The real difference between the necessary being of God and the contingent, created finitude of a human being is the difference between (i) and (ii).]
1D, 2D, 3D = spatial composite (line, width, height)
4D, 5D, 6D = temporal illusory (length, breadth, depth)
7D, 8D, 9D = spectra energies (continuous, emission, absorption)
Symmetry/entanglement:
1D, 4D, 7D line, length, continuous
2D, 5D, 8D width, breadth, emission
3D, 6D, 9D height, depth, absorption
Conclusion: Humanity needs to immediately swap from "Newton" to "Leibniz".
Our calculus is incorrect (Leibniz > Newton):
What is the difference between Newton and Leibniz calculus?
Newton's calculus is about functions.
Leibniz's calculus is about relations defined by constraints.
In Newton's calculus, there is (what would now be called) a limit built into every operation.
In Leibniz's calculus, the limit is a separate operation.
Our Universal Constants have convoluted answers. Leibniz's Law of Sufficient Reason fixes this in a day.
(FUNDAMENTALS > specifics)
😅😅😅😅well information good show you 😅
PRINGLES
This is the most sensible comment in the entire chat section 🥲🥲