The Maths of General Relativity (6/8) - Energy fluxes 

Cherish it while it lasts

well, you pay for internet

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​@@elisei4ik ❤

he wanted to click on "like", but suddenly the spacetime got disturbed

@@mahanstyle376 I don't know about spacetime, but those unliker brains, surely got disturbed.

bots

@@carlosgarcia3341 this video series is about spacetime, and how it is bended by energy

@@mahanstyle376 well, my comment is about the bent minds of some people.

amazing !

@@DaRios_Tristan Yep, great choice of sound. It doesn't irritate ... too much ... for me ;-)

Glad you like it :)

@@ScienceClicEN I set reminders for each videos premiere. So far they have been released on a schedule that matches the amount of free time I have to study things unreleased to my course. I don't know if this was intentional, but it was helpful.

what is the unit of temporal speed??

@@hassanhachem8764 Rate of change of time in the current reference frame, with respect to proper time of the object you are tracking (in its own reference frame). The units are then time per unit spacetime.

@@oatmongen4263 plz .can u explain more ur answer . thank u a lot

Update: it is helping

Its never too late!!

You can always studyy physsics, physics isn't a profesionao degree its only knowledge, it never should be late to obtain more knowledge, go for it! If you don't have time to go to college just take 2 books and see how it goes!

Guiablo Bravo

Thanks for the kind words guys. Definitely never too late, especially when it comes to learning!

Thank you very much for the support 🙏

Can be

I understand, I was a bit worried about this. I chose this way because it's easier for taking advantage of the longer side of the 16:9 video format. Also I thought it might be more intuitive for people who never saw spacetime diagrams, because this is how we perceive time when reading, from left to right.

Yes, but this is a good explanation

What is ill defined in GR is the total energy contained within a "slice" of spacetime. This can only be defined for certain spacetimes (namely those with time invariance at infinity, or more technically with a Killing vector field, an isommetry, which is timelike at null infinity for an asymptotically flat spacetime) But if you consider only one object and its associated energy-momentum tensor then it is well defined, its "energy" is in a way simply its mass, which propagates through the dimensions of space and time. Or rather what we usually call the "energy" is the time component of the momentum of the object.

We have to recognize our Awareness is a computation/input/output Gate in a quantum system. We Space/Time is the Answer.. We are trying to understand the question.... Or Unknown Invariants ?

@@ScienceClicEN The only thing I understood that I need to study further. No problem. Just I might poke with more stupid questions like this. Here's one already. As far as I know, the inertial mass is an emergent phenomenon that results from confinement, imposed by strong nuclear interaction on the gluons in the nucleus or by the Higgs field. I know this thing mainly has to do with momentum transfer. But this is also how E=mc^2 is derived. So what is that with the pov Emmy Noether left us with?

@@smileifyoudontexist6320 No probably didn't grow such awareness. Just self-studying the math underlying QM and GR so that I can actually understand them.

@@aniksamiurrahman6365 ​ If it didnt grow? what size were you 100 years Ago? Yes True. When all or enough of the Math is understood.. One will say "Something very strange is happening!" To think of how would SpaceTime Twist and curl and bend on our local scales? In relation to Individual world lines. In relation to Celestial reference ponts, and then to Cosmic Reference Points . considor the actual Topographic transformations Its probably 1 singular worldLine this planet. Surrounded by electromagnetic field. When a spaceship enters the atmosphere its like cutting into the sheath of a wire. Thats a hint to the nature of quantum Cosmology?maybe

In physics it's quite hard to say that whether something "exists" or not. In a way you could even argue that the word "exist" does not really make sense. "Time", just like "space" is a concept invented to describe our world. Just like you need 3 numbers to describe the position of something, you also need 1 number to describe *when* an event takes place. This is crucial, you cannot describe the universe without this 4th parameter. It's this parameter that we (usually) call time in physics. In other words its not something that "exists" or not, but simply a necessary ingredient in the descriptions that model our world. On the other hand, if you are talking about the "passing of time", asking the question of whether of not time really "goes by", then the standard answer according to modern physics (in my opinion) would be that the passing of time is simply something that we experience, a perception issue, it can be related to the increase in entropy inside our brains for example. But "outside our head" it would not really make sense to talk about the "passing of time". In particular in general relativity we consider the universe as a whole 4 dimensional object, in which the "past" and the "future" coexist. We are simply experiencing successive slices of this geometry, hence our impression of the "passing" of time.

Excuse my eng. Suppose that time is real. In time-space fabric it is assumed to be one of the two axes. For us human, it is a measure of our life. Ett it is an "time areal"density (volume density = E/m3). Time areal density variations =dE/d(tt) are in fact a measure of steady or unsteady state, i think. In one dimension, if dE/dt = 0 it is a steady state procces. If dE/dt < 0, or dE/dt > 0 it is a unstady state proc.

@@colfrancis9725 In some sense, the question about determinism has already been answered by General Relativity. The Past and Future are real and exist, and therefore the universe is deterministic. Now, don't get me wrong! Deterministic is not the same thing as predictable. We can know for certain that "something" will happen in the future but we cannot predict exactly what that something is. We can place limits, such as no FTL causality, but we cannot make accurate predictions without perfect particle knowledge and that is forbidden in Quantum Mechanics. Free Will can still be saved by attributing some aspect of consciousness to this fundamentally unpredictable quantum mechanics.

Yes, exactly like Continuum Mechanics

I think more fundamentally than even energy is information, with energy being a particular manifestation of it, Mass being a particular manifestation of it, and all the various quality of anything we can experience being nothing more than information encoded within various aspects of our reality. This was the big understanding from "Maxwell's Demon" thought experiment, with energy being able to be put in what entropy would consider a disordered state, so long as information is still following entropic potential.

It is counter-intuitive indeed ! The "inflaton" field is a postulate for describing inflation, but dark energy is described by the "cosmological constant" which is not the same thing. It is a constant that we can add to Einstein's equation without breaking the theory. This constant can either be interpreted as a "background curvature" for spacetime, or as a "dark energy" (whether we put it on the left side or on the right side of the Einstein equations). The energy-momentum tensor associated with dark energy is simply proportional to the metric, which means that this is completely isotropic and homogeneous.

@@ScienceClicEN Yes, Einstein was right, even when he thought he was wrong. That being said, I've seen presentations where the potential of the inflaton field was much higher in the past (causing rapid inflation in the early universe) and then a phase change, or tunneling to a lower potential of the order of 10^-120 occured (ie the current cosmological constant), implying that dark energy is the same field as the inflaton field, just at a much lower potential, another case of a spontaneously broken symmetry. As I see it, it's just the quantum-mechanical explanation of the same phenomenon. Dark energy exists, therefore it must have a quantum (the inflaton) and a quantum field (the inflaton field)

Have you had experience using 3d vectors in Newtonian/classical physics?

A proton has *rest* *mass,* so it can't be accelerated to *c.* It requires an infinite amount of energy. It could be an *active* *mass* and source of spacetime curvature and a gravitational force, but the *passive* *mass;* e.g. if it's a simple hydrogen atom, the electron would be dominated by the effects of the electromagnetic force and binding energy between the nucleus (proton + neutron) and the electron. The particle colliders conduct experiments accelerating protons near *c* to study the collision. Gravity is too weak on those scales, so a better example of local energy and momentum density/flux creating more gravity would be to add energy to a gravitational system. In the earth-moon system, if energy were added to increase the rotational velocity of the earth and everything else remained constant, the increase in angular momentum and energy density would curve more spacetime. The opposite over time is true because as the earth slows in rotation the gravitational binding energy between the earth and the moon will decrease.

Did you mean like transforming the stress-energy tensor to spherical coordinates?

I don't know. If you find out, please write a comment here. Rough ideas: Go from the basic, intuitive, definition of the stress-energy tensor. Off-diagonal components represent a flux of the x-component of momentum in the y-direction, which is often considered as something like a shear term for a viscous fluid. So, for an ideal fluid all off-diagonals certainly are 0. If we model the content of spacetime as a fluid (which is often done), your question is then effectively asking - can we find co-ordinates in which the fluid behaves as an ideal fluid. I suspect the asnwer is NO, since viscosity seems to be an intrinsic property of the fluid and not of the co-ordinates. But I don't know. There should also be some unusual configurations of space which will effectively prevent transfers of momentum in certain directions anyway. Consider, for example, a spacetime which has a degenerate y-co-ordinate. Let's say that all matter is confined to a finite set of y-co-ordinate values. The situation is then like forcing all matter to flow through narrow pipes at fixed y-values, keeping those pipes narrow (reaching 0 width in the limit) effectively prevents any transfer of momentum in the y-direction. Anyway, why do you want the stress-energy tensor to be diagonalised?

Beware these are two different tensors : On the one hand you have the energy-momentum tensor, which contains in the upper left corner the "energy density". This tensor is zero for the Minkowski spacetime, because it describes an empty universe. On the other hand you have the metric tensor, which describes the geometry of spacetime. This is what you were refering to. There is actually a choice of convention in General Relativity, in the metric tensor we can choose to put the negative sign either in front of the time component, or in front of the 3 spatial components. In my series I decided to go with the convention that time is positive, and space negative, but you will also find the opposite convention quite often.

​@@ScienceClicEN thanks for the reply. That clears up my confusion, and gives me more questions. Looking back at the Metric Tensor wiki I do now see where it eventually is related to the stress-energy tensor... and now I'm sorta understanding the old saying 'Space-time tells matter how to move; matter tells space-time how to curve.'

Good question, there is no preferred frame of reference. The power of general relativity is that we can choose any frame we want (and even any coordinate system, which is a more general choice), and it will always be consistent.

Indeed, you don't need a rectangular coordinate system (you can choose whichever you prefer) but it was simpler for the visuals. What is described by the energy-momentum tensor of *a single object* is the flux of momentum through each surface corresponding to a coordinate on the grid. This is therefore a collection of vectors which are all proportional (for a single object) to the object's velocity vector. However for multiple objects, or for a "fluid", you sum the energy-momentum tensors of everything, and therefore it becomes more complex. In a way this is closely related to statistical mechanics and how we describe fluids. This is why we try to give an intuitive understanding for each term in the tensor, as density, pressure, momentum, and viscosity.

@@ScienceClicEN Oh, so the energy-momentum tensor does not describe a single (velocity) vector, but helps compute something else based on it? In that case, where do we get the values in the matrix? Are there different types of energy such that their velocity affects spacetime differently? (Or just one type but it's filled with constants in a convenient form?) I guess I was confused since the prior videos described where their numbers came from and I assumed energy only came in one form, so it would be fully described by the velocity vector

For a single particle / point-like object you can compute its energy-momentum tensor only with the components of its velocity, and its mass "m". It's given by : T_mu_nu = m * v_mu * v_nu, with a Dirac function such that the tensor vanishes everywhere other than on the object's worldline. However the idea is that the total energy-momentum is usually the sum of the energy-momentum tensors of many objects moving in all directions (think of a gas for example). Also we can also define the energy-momentum tensor for the electromagnetic field, or for dark energy for example, so it's not always defined using the velocity of a point-like object.

@@ScienceClicEN Ok, I get it now. The components of the matrix in the example are (easily computed from) the worldline, but you're interpreting the components more broadly to prepare us for additional forms of energy, like a point within an EM field, or a combination of sources. Thanks! I'm looking forward to the rest of the series.

Exactly ! I hope you'll like it :)

;)

Edit: this is wrong. I'm not fully understanding it myself, but I think the 00 component is associated with rest mass and the 0i / i0 components ("momentum components") are associated with the kinetic energy or the momentum of the object. Also as he said in the video the tensor is *not* invariant under coordinate transformations, so the motion (and thus the energy/momentum) of a massive object is dependent on the coordinate system.

@@sherlock_norris A tensor _has_ to be invariant under coordinate transformations by definition. The components may change, but the object they represent must not change. According to wikipedia the 00 component is relativistic mass density, which contains all energy including kinetic. en.wikipedia.org/wiki/Stress%E2%80%93energy_tensor#Identifying_the_components_of_the_tensor So what i think this means is that while observers may disagree on the numbers, they will agree on the consequences(like the fact that there is, or is not, a black hole. You can't create a black hole transforming to a higher velicity observer). I'm just not quite seeing how that works out.

The 00 component for a point-like object is its "total" energy density, so it also includes its kinetic energy. Actually in GR you don't really have a clear separation between the different kinds of energy (mass / gravitational / kinetic). They all come from a same term. But you could interpret that : - the rest mass energy comes from the inertia of the object through spacetime - the kinetic energy comes from special relativity time dilation (as the "temporal velocity" of an object changes when it moves through space, so the 00 term of its energy-momentum tensor is affected) - the gravitationnal energy comes from the "curvature of time" (since the 00 term of the metric also affects how the temporal velocity of the object changes) The tensor is Lorentz invariant, in the sense that it's a geometrical object in spacetime. But the interpretation of its components may vary : something could be said to be "energy" in one frame, but "momentum" in another frame. It's very much like the electric and magnetic fields. In one frame you may have an electric field, and in another frame this same field might be interpreted as a magnetic field. Also, note that *all* components of the energy-momentum tensor will affect the curvature of spacetime, not just the 00 term.

@@narfwhals7843 Ah, so I have made the classic mistake of confusing the tensor with its matrix representation. I guess I can't answer your question myself then, but I found a video ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-UUeWsfsJSX0.html that goes much deeper into the math than this one, especially regarding the transformation of the tensor under coordinate transform. I guess while these ScienceClick videos give a great surface level/intuitive understanding, the maths beneath it stays hard...

@@ScienceClicEN "Also, note that all components of the energy-momentum tensor will affect the curvature of spacetime" That is my point. I'm having trouble understanding how two observers can see the same curvature when they disagree on the components. But they don't have to see the same curvature, do they? They can disagree on which dimension is which as long as it leads to the same result. Thanks for your reply!

It might be a bit unusual but I actually never used books in my academic background, I always preferred watching explanatory videos or online lectures. That's how I first learned GR (watching the lecture videos from the amazing Richard Taillet, which are in French) just before making this series. However I did recently use the lecture notes from David Tong who was my GR professor last year, they are really good, but this is already quite advanced : www.damtp.cam.ac.uk/user/tong/gr/gr.pdf

@@ScienceClicEN That's great ! Thank you so much.

@@ScienceClicEN Which software do you use to make animations in your videos..?

I think you might like this book - "A Most Incomprehensible Thing: Notes Towards a Very Gentle Introduction to the Mathematics of Relativity" by Peter Collier. Start reading it for free: a.co/3NcCkk4 The book starts easy, and goes all the way.

But you need the physical book, it doesn't work on Kindle.

Glad you like the series ! I'd say not to worry about the position of indices. The only thing that matters is that when you write an equation, the indices that are present on both sides must be either both up, or both down. But otherwise you can always "raise" or "lower" indices, using the metric tensor. For example T_mu_nu (with indices downstairs) is g_mu_alpha*T^alpha_nu (so now you have T with one index up and one down) which is also g_mu_alpha*g_nu_beta*T^alpha^beta (which is now T with both indices upstairs)

I asked basically the same question and here is his answer ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-JKQMre-bze4.html&lc=UgyVajCNfDxpnTuiD0J4AaABAg.9HrGGBI3A-k9HrNu30vdjJ A tensor, by definition, can't be relative. But its components can be as long as they represent the same object. The curvature is also a tensor, so all observers will agree on how those two tensors relate to each other, even if they don't agree on their components. That is part of the power of a tensor equation.

@@narfwhals7843 Thanks! The electricity/magnetism analogy helped me grasp the idea of different components appearing as different things to different observers.

In a way it's both. The energy which is described by the energy-momentum tensor is the energy of all the objects that the universe contains. So if the universe contains only an apple, the energy contained in the universe is the energy of the apple. But if the universe contains multiple objects, the purple stuff represents the total energy of all these objects combined. For example you might have an electromagnetic field which fills your spacetime. In this case, there would be purple everywhere on the grid, with more or less intensity depending on whether or not the EM field has a lot of energy at each point.

@@ScienceClicEN Firstly: thanks for that extensive answer! So, just that I got it right: If it is also the apple, then it would be maybe more correct (but less instructive) to make that box fully purple as the Apple fills basically the whole box. If it would for example get shot by an arrow right through, it would get even more purple (higher energy density and also momentum density). Adding an electromagnetic field would make it _even more_ purple than the apple itself. Did I get this right? Thanks a ton, you saved my life man.

Yes basically you can just think of this as a fluid, like water that we would see from the top. And the more "purple" it gets, it means the more energy/mass/objects is passing through the point. Inside the Sun for example, the plasma of hydrogen which constitues the star is made up of many atoms, but to describe it we "sum" all the atoms together, thinking of them as a fluid, with a density (which is greater when more atoms are stacked into a same volume), a pressure, a momentum, etc

@@ScienceClicEN Alright, I got it! So even the apple contributes! I was first confused because it would "block" other massive objects from entering the box, leading me to the false idea that if the box is inside the apple, the energy density would remain constant as long as the apple persists. But you had a point with the energy of the electromagnetic field for example which disproves this:) Man, you're a really good guy, I searched for something like that for days (literally). I'll become a patron right away, you deserve it! Keep it up, you're helping me and others more than you can imagine ❤️

Thanks for the support I am very glad you like the series !

Ahah have you tried running the regression on the original series on the French channel ? I wonder if it would be the same curve

Assuming no physics backgrund before these brilliant videos, and a self-study approach, I'd recommend Leonard Susskind's "The Theoretical Minimum" series - Book 1 Classical Mechanics then Book 3 Special Relativity and Classical Field Theory (both absolutely essential prerequisites). You don't need the Quantum Mechanics book. These books are pitched at a perfect level, they don't skip any mathematics but they have tons of explanations. Then to start on GR, I don't think MTW Gravitation is ideal, probably a friendlier book like Carroll "Spacetime and Geometry" or D'Inverno. Unfortunately even these are hard going and you'd ideally want a friendly GR tutor to answer the inevitable million questions. Unfortunately I don't think anyone's yet written a friendly book on GR - Susskind has vague plans to do it one day. Maybe ScienceClic Alessandro could do it? He's probably already done it in French! French is easier to learn than GR...

Incidentally, Sean Carroll also has some amazing videos on GR in the "Biggest Ideas in the Universe" series on RU-vid.

@@JoshuaSmithNZ Thankyou for your reply! this is really helpful I have those books on my list and will surely get them now and it's a good thing I'm learning french then...

@planet42 Something to work towards then

Yes, the Ricci tensor component takes input coordinate positions and outputs the curvature at those positions.

Exactly

Very an interesting topic yet very arguable 👍🍵🐙

Yes, the viscosity is momentum flux and shear stress. because, shear stress contains Txy Txz Tyz, and momentum flux are Tyx Tzx Tzy . they're equivalence, which means, the component ended up to be the same by the symmetry.

The energy-momentum tensor is the flux of 4-momentum, so it is indeed the flux of a vector field. The 4-momentum can be considered to be the product of the energy density with the 4-velocity of the energy. Basically it's like the flux of 4 scalar fields, one for each component of the vector field (which is why it makes a 4x4 matrix, instead of just a 4-vector)

@@ScienceClicEN OH ok, Thank you so much!!

He should have said "cement inside the sun."

I agree it's quite hard to get an intuition for it. Personally I think of it this way : the i,j component tells us how much motion in the i direction gets "transferred" to points nearby in the j direction. For example, the x,x component tells us how much spatial motion along x gets transferred in the same direction : it's pressure, telling us how matter "pushes" (i.e. transfers its motion) in the x direction. The x,y component tells us how motion along x is transferred to points nearby in the y direction : this is viscosity, transverse transmission of motion. The time components can be interpreted in the same way : The t,x term tells us how much motion through time (i.e. how much energy) gets transmitted to points nearby in the x direction. This is momentum, it tells us how energy "moves" in the x direction.

@@ScienceClicEN Yeah, I rewatched a few times and googled abit for more info and think I am starting to understand the logic. I was especially impressed with how you made an analogy of x, y component with viscosity, it is brilliant and it contributed the most to me understanding it. And x, x component really was the most difficult, but even that started to click with a few rewatches. I still don't have it completely understood, but your video helped. Also, thanks for making this series, it is fantastic and I really hoped all professors explained and motivated ideas like this before jumping into mathematics and abstractions. If you aren't already, you would be a tremendous teacher/professor.

Next tuesday (the 5th I think) ! And the final one will be on tuesday the week after

@@ScienceClicEN Nice! :) Can't wait :D!

That’s exactly what it mean to be moving through time at the speed of light. The rate of change caused by individual particles inside the collection is maximized. If the collection begin to move through space, this rate of change slows down, and that’s what we called time dilation.