You helped de-mystify the metric concept, and your derivations showed algebraic tricks with manipulating indeces that I’d not seen before. Seeing a wave equation come out of the Einstein equations with the speed of light helped me understand their connection.
phenomenal video as always, and incidentally perfect timing with the little explanation on gauges and gauge transformations. i just recently i decided to start reading papers on Gauge Theory Gravity, a flat spacetime formulation of General Relativity using Clifford Algebra based Geometric Algebra/Calculus and new gauge-related axioms to constrain the field equations. Up until now, i was having a hard time understanding why transforming from flat spacetime to another flat tangent spacetime at some point would matter at all, but i think you tipped me off into the right direction. again, thank you for all the fantastic videos.
Ist ja lustig, habe deine Videos zur Abiturvorbereitung geschaut und schreibe in 2 Wochen auch die GR Klausur. Ebenfalls meine letzte Klausur für den Physik Bachelor.
Hello I just wanted to thank you so f*ing much, I spent 2 days on this part of my General Relativity exam and because the material provided was not so explicit I couldn't figure out some parts. You literally answered multiple question I asked myself while studying, this is an indication of an EXCELLENT explaination, clear in every part and especially in calculations, which are always lacking in explaination in this part of physics, even in "sacred books" such as Wald or Carroll
Dear EigenChris, very good.. thanks! Am finally starting to understand 4D-SpaceyTime Tensor/Up_Down_Index notation! And thanks for clarifying the Lorenz/Lorentz confusion, which may be a kind of cosmic joke.
I'm always uncomfortable with 'ignoring' very small terms but then carrying on using an equals sign. Surely we no longer have equality? Or is there some underlying taking of limits occuring?
It makes me uncomfortable too. I think there is a formal way of handling the small terms called "perturbation theory", but I haven't learned it properly. If you like, you can think of the true answer being a Taylor series, and we're just taking the 0th and 1st order terms and ignoring the rest. Similar to how many things in physics are modelled as simple harmonic oscillator (2nd order), we're modelling changes to the metric linearly (1st order).
Yes there is an underlying limit operation. One can, and formally should, keep those small terms around during the manipulations. One just has to keep track of their magnitudes during all the steps (often it is not necessary to calculate them explicitly, just to obtain bounds for their magnitude). Then in the end one has a true equality and some "ugly" corrections. But for those we know how big they can be relative to the original perturbation (displacement). They should be orders of magnitude smaller, at which point we may just want to drop them. Often some technical assumptions must be added to make it all rigorous.
You also have to remember that the small terms would be so physical small that in the case for GR, they might be smaller than Planck length meaning they "possibly" would be completely universally insignificant. So 0 is acceptable, that at least how I mentally justify it to myself.
On 11:10 we have derivatives with ~ acting on the metric h. In order to write them as ordinary derivatives I get an extra term of h_{αβ;κ} =h_{αβ,κ} +h_{αβ,γ} ξ^{γ}_{δ}(partial x^{δ})/(partial x ~^{κ}). I've used ; to denote derivative with~ and , to denote normal derivatives. Basically metric derivative wrt ~ = metric derivative wrt normal coordinates+ terms that involve ξ * the inverse Jacobian* the derivative of the metric wrt to normal coordinates. In order to make this term vanish you must assume that the derivative of h wrt normal coordinates is very small which is plausible for sure, but I'm not sure if we've assumed it. I tried to do this myself, got stuck there and then decided to watch your video, but I think you didn't address it. Is this an extra assumption we must take?
I have gone through your Tensors for Beginners and Tensor Calculus videos which are excellent. This series is also monumental. The explanations are clear and complete. If only some professors could do so good a job! Question regarding gravitational waves: You have demonstrated that the Einstein Field Equations reduce to a wave equation under the Lorenz Gauge transformation, so in a vacuum, where the energy-momentum tensor is zero, waves in space time are propagated the same as EM waves in space. In the case of EM waves, the wave is initiated by a boundary condition like an alternating current in a wire. In the case of two enormous masses (black holes) orbiting each other, what does the Energy Momentum Tensor look like? Are the terms sinusoidal functions of time?? Your insight would be appreciated.
Hi Chris, Thanks from the bottom of my heart for the wonderful lectures! I am enjoying each bit. @14.42 we have a recipe for the displacement field as a solution of another wave equation. My questions are follows 1. How do we know the solution for the displacement field exists? 2. Even if we know the solution exists it is still very vague for me to be able to find the solution. Do you use a Green's operator? 3. What will be the initial conditions for such equation ?
I think this is the first time in your videos that you explicitly deal with a displacement as a coordinate transformation. I always understood the coordinate transformations you used to basically be linear maps. And I actually made the assumption that tensors are only invariant under those, and not those that change the origin, but I see why that would be wrong. But is there something else we have to pay attention to when we use something other than a linear map as a transformation? Or is that just trivial?
You can use any coordinate transformation you like. The basis vectors are just the tangent vectors along the coordinate curves, and tensors will transform according to the jacobian and inverse jacobian matrices. I think I already showed this in the 105 videos when we dealt with Rindler coordinates. Tensors live in the "tangent space" attached to a given point on a manifold. Changing coordinates of the manifold only changes the basis of the tangent space, it doesn't change any actual geometry.
This was great! I have a question regarding the approximations made to get to this point. So first, you walked us through a derivation of the linearized Einstein equations, which is a perfectly sensible thing to do and very common. But I'm confused about the derivation of the Lorenz gauge in linearized gravity, since it seems like that's only an approximate gauge, at least as you walked through here, since you made assumptions about terms being small. This confused me since it seems like even in linearized gravity, the Lorenz gauge is an approximation, and possibly not satisfied exactly by any real transformation. Is that true?
Sorry I must have seen the notification for this comment, but forgot to reply. It's true that everything in linearized gravity is an approximation. The real Einstein Field Equations are non-linear. But the "linearized" approximation is considered "good enough" when we are dealing with spacetime curvature that's relatively weak (gravitational waves from distant sources are very weak). The non-linear effects are assumed to be small enough for gravitational waves that they can be largely ignored.
Is there a concrete simple example of a Lorenz Gauge frame? I understand that it's just a simple way of writing the field equations in a way that makes it apparent that it is a wave equation, but I would like to picture what kind frames satisfy the Lorenz Gauge.
In the next couple videos, we'll see a specific example of a Lorenz gauge called the Transverse-Traceless gauge. We'll use the metric equations that pop out of that to get formulas gravitational waves. I honestly haven't calculated specific Lorenz gauges except this one.
Thank you so much! I might've missed something -- how can we choose xi such that the partial derivative of h w.r.t. beta is the d'Alembert Operator (slide 31)?
Thank you Cris, for another great video and explanation! I have a question about the gauge invarience in GR. The gauge invarience in the EM filed results in charge conservation. What is the quantity conserved by the gauge invarience in ( curvature?) the Riemann Tensor?
I'm not sure I'm knowledgeable enough about gauge theories to answer this properly. I can maybe say a few things that you can use as a starting point to google more. I know that in QFT, we normally view spacetime as having an "internal symmetry group" at each point (e.g. U(1) for QED). Taken together, spacetime (called the "base space") and the internal symmetry groups (called the "fibers") are a mathematical structure is called a "fiber bundle". A choice of gauge in in QFT is the same thing as making a choice of "connection" between the fibers (this choice of connection immediately leads to a choice of covariant derivative). In GR we don't really "choose a connection" or "covariant derivative" like this... there's only one connection/covariant derivative we every use, which is the Levi-Civita connection. So I'm not sure if the gauge invariance in GR is completely analogous to the gauge invariance in QFT, and I'm not sure if we end up with something analogous to a conserved charge. This is pretty out of my depth though, so I can't really say one way or the other.
I only understood about 10% of this, but even so I think I got the punch line. By choosing the right gauge the equations simplify so it is much easier to understand, and although the answer is not perfect, it is close enough for government work.
@@eigenchris It is mostly due to having a bad memory in that I watched many of your videos a while back but then promptly forget what I learned. My other problem is while the notation you use is quite elegant, it is hard to keep track of what all those indices mean. There may be a way to help fix that but would probably require some sort of tool that could link an index to a graphical (as opposed to symbolic) representation of it. Thus say just in a x1, x2, x3 vector say one clicked on the 1 of the x1 it would draw a line from that to the X1 axis and if one clicked on the X of the X1 it would draw a line form that x to the vector X1 along the X1 axis sort of like a tool tip or like a here you are at the mall map. Thus one could click on any symbol and have a line drawn from it to a diagram that shows what that symbol represents. Unfortunately, as far as I can tell, no such tool exits, but it would be cool if it did. Interestingly enough, I just made a RUclips video that sort of illustrates this basic idea, where those lines I mention above are what I call yarn that go with what I am calling bookmarks where the yarn is sort of like the yarn one sees in a detective show where they have a forensic board with pushpins and yarn to make connections. Here is a link to my video that shows this (which albeit I sort of messy in that my handwriting is as bad as my memory. ruclips.net/video/TsneUuzapIY/видео.html And here is a link that shows a forensic board (or murder board) that illustrate what I mean by yarn. Thus a bookmark is the virtual way of making connections like one does on a murder board, but instead of photos of suspects and the like, one could links indices to drawings using it. ruclips.net/video/7IrBQFmAUi0/видео.html With an aid such as that I might be able to connect the dots a bit better between the notation and what that notation denotes without having to rely so much on my (bad) memory.
Try doing exercices, there's only a limit of what we can learn just by listening to someone, no matter how well they explain. I learnt it the hard way at university, by realising that I could ingest an immense amount of lecture, but that I would only begin to truly understand (and remember it) when doing the exercices (I have the chance to have university that put a big emphasize on exercices and I was surprised to see it's not so everywhere).
I dont know the answer either but if the waves could and there would be a lens effect, would it then not be possible to obtain information from inside the hole?
@@imaginingPhysics Yes that's one of the many paradoxal consequences that crossed my mind, not sure what it would be. It also raises the question if even normal gravity (not wave) could be affected (even very slightly) by any body on its path.
Any insight how we might interprete Lorenz gauge cond.? Its a "continuity equation", right. For the EM 4 potential it can be interpreted as choosing coordinates where the "scalar potential (the zeroth component) is conserved". This can be seen by integrating the divergence equation 21:55 and using divergence theorem. So, could this metric Lorenz gauge be interpreted as choosing a frame where "the metric is conserved" in the sense that the contractions cannot just vanish or appear out of nowhere? Metric must propagate if it wants to change. This would make sense if we want to see Gwaves?
The "continuity equation" makes sense for vectors--it just means whatever goes in a box must leave. The continuity equation for the metric perturbation, however, I'm not sure how to make meaning of. Can you explain what you mean when you say "contractions cannot just vanish or appear out of nowhere"?
@@eigenchris Ill try. Generally, if you move at x direction, g_xx tells you how much your x-coordinate changes. But your x-coordinate can also change if you move at y direction; g_xy gives gives you that rate. And g_xt tells you how your x-coordinate changes with time. Right? So we have "a set of meter sticks" g_xt, g_xx, g_xy, g_xz. Same goes for inverse metric. Now I think we can form a vector (g^xt, g^xx, g^xy, g^xz). It is not a 4 vector, but its not meant to be. We do want it to change, and we are looking for a frame where its divergence g^xt,t + g^xx,x + g^xy,y + g^xz,z is zero. And what does it mean when its divergence is zero? I think it means that if meter stick g^xt is stretching, some other meter stick must be contracting, and vice versa. It also means, I think, that local changes in metersticks cannot spring out of nowhere; there had to be fluctuations going on near by.
@@imaginingPhysics That's a very interesting interesting way of looking at it. And indeed, with a "plus"-polarized GR wave, any "extension" in the x-direction needs to be "stolen" from the y-direction, where a contraction happens. I can see that making sense. The only missing link for me is this sentence: "But your x-coordinate can also change if you move at y direction; g_xy gives gives you that rate". I normally think of g_xy as measuring the angle between the x and y basis vectors. But maybe the two concepts can be made to be the same?
@@eigenchris Yeah g_uv as an angle is more accurate interpretation. My language was sloppy and incorrect as I tried to explain the idea: Of course if one moves at y direction then by definition only the y-coordinate changes. Still, if x and y axes are not orthogonal, then there clearly is a sense in which one seems to going in x-direction also (draw a picture). And this is what the metric component g_xy measures (because if represents the angle). At the moment I don't find better words for it. Covectors ("stacks") might be useful too.
I'm not sure if I was worried about covariance/contravariance when I wrote those. The formulas at 20:05 are just the standard formulas from a non-relativistic E&M class. If you're going to do E&M in a way that's Lorentz-invariant, E and B fields are the wrong way to think of things. You need to think in terms of the Faraday tensor at 20:45, which is a twice-covariant tensor.
(Sorry, tried Mathtype paste here -no joy) ...have you ever thought of (x alpha) and (x alpha prime) as yr x co ords instead of (x) & (x tilde)...it greatly simplifies comma=derivative notation totally avoiding ambiguity
I started using x-tilde in all my videos because i found it easier to read than the tiny prime symbol. In this video it makes things abit worse, but I stuck with it for consistency.
@@eigenchris Totally understandable! BTW just an acknowledgement of the great work you've done. Esp yr analogies on covectors (sthing I always struggled with)& of course the SR & GR work. Have you ever read Quanta magazine? A little above New Scientist but good on Quantum Gravity & related stuff. Another interest is Unified Field Theory (Mostly above my head!)but there are 2 excellent 100 page papers on its history up to I think the 1970s.(online)...can dig them out if you're interested.
When one contracts the wave equation for h_bar with the Minkowski metric, one gets an equation that says that the d'Alembertian of the trace of h is zero. Can I plug it in in the first wave equation to simplify it to a second wave equation which simply says that the d'Alembertian of the tensor h is zero (as opposed to the full tensor h_bar with the sum of the trace)?
In the next video I plan on doing something along these lines... although I haven't fully thought it through yet. I'm not sure if you get that for free in the Lorenz Gauge or if you need to pick a special coordinate system.
@@eigenchris I think you get that for free in the Lorenz gauge and another way to see that I think is to consider the Einstein Field Equations in the vacuum, and they reduce to the Ricci tensor being 0 (the space is Ricci flat), and one can do the same calculations done in your vids, imposing the Lorenz gauge afterwards to get the wave equation simply for the disturbance in the metric h. Or am I making this up? :P
@@tommasoantonelli7176 Yes, that does seem to make sense. I think I was getting confused because in the next video, I plan to set the trace of "h" itself to zero. But in an arbitrary Lorenz gauge, even though "trace of h" isn't zero, the "d'Alembertian of the trace of h" does to go to zero based on the reasoning you gave.
So we chose a coordinate system that obeys a wave equation to get a metric that obeys a wave equation. How can we be sure that the waves this predicts are really a feature of the metric and not a coordinate artifact? Isn't that why Einstein changed his mind about gravitational waves like 15 times?
In 109e we'll look at how the proper distance changes between parallel geodesics, and we'll see it vibrates along with the wave. This is a result that doesn't depend on coordinates, and it basically means free particles will follow the animations I show in this video. It's a good question though.
@@eigenchris thanks, looking forward to it. Will you also talk about what the fact that it moves particles means for the energy of the wave and what this means for gravitational energy as opposed to stress energy? Or is that outside of the mathematical scope of this series? I've been wondering if there's a way to get an energy tensor that includes gravitational energy.
@@narfwhals7843 I don't really understand the mathematics of how gravitational waves carry energy yet, and I wasn't planning on making a video on it on my first run. After all the main videos are done I might make a GR video once in a while to answer questions like this.
@@eigenchris Actually, I have to be sorry for my comment. It's easy watching from the outside to criticize glitches in a video or explanation. Were we should focus is on the content and usefulness of the explanation or information content. My apologies.
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I loved the little discussion you had about gauge transformations in E&M. I’d absolutely watch a video on relativistic E&M
I'll probably make a video on that later this year... make it an optional video at the end of the 104 videos. I want to finish GR topics first.
I really appreciate your lectures/videos. This one answered a bunch of long-standing confusions.
Thanks! I'm curious: what did it clear up specifically?
You helped de-mystify the metric concept, and your derivations showed algebraic tricks with manipulating indeces that I’d not seen before. Seeing a wave equation come out of the Einstein equations with the speed of light helped me understand their connection.
phenomenal video as always, and incidentally perfect timing with the little explanation on gauges and gauge transformations. i just recently i decided to start reading papers on Gauge Theory Gravity, a flat spacetime formulation of General Relativity using Clifford Algebra based Geometric Algebra/Calculus and new gauge-related axioms to constrain the field equations. Up until now, i was having a hard time understanding why transforming from flat spacetime to another flat tangent spacetime at some point would matter at all, but i think you tipped me off into the right direction. again, thank you for all the fantastic videos.
Thank you, nice preparation for my last exam 🤩
What exam?
Ist ja lustig, habe deine Videos zur Abiturvorbereitung geschaut und schreibe in 2 Wochen auch die GR Klausur. Ebenfalls meine letzte Klausur für den Physik Bachelor.
Hey man, i dont watch this channel a lot but im really impressed and appreciate that you just keep posting no matter what. Thank you
It's always a pleasure to follow your videos, I'm really thankful for your effort to explain complex subjects in the plainest way.
no professor I saw teaches this detailed. Thanks a lot.
Hello I just wanted to thank you so f*ing much, I spent 2 days on this part of my General Relativity exam and because the material provided was not so explicit I couldn't figure out some parts. You literally answered multiple question I asked myself while studying, this is an indication of an EXCELLENT explaination, clear in every part and especially in calculations, which are always lacking in explaination in this part of physics, even in "sacred books" such as Wald or Carroll
Dear EigenChris, very good.. thanks! Am finally starting to understand 4D-SpaceyTime Tensor/Up_Down_Index notation! And thanks for clarifying the Lorenz/Lorentz confusion, which may be a kind of cosmic joke.
Thanks a million for these wonderful explanations.
Awesome maestro 👏🏻
I'm always uncomfortable with 'ignoring' very small terms but then carrying on using an equals sign. Surely we no longer have equality? Or is there some underlying taking of limits occuring?
It makes me uncomfortable too. I think there is a formal way of handling the small terms called "perturbation theory", but I haven't learned it properly. If you like, you can think of the true answer being a Taylor series, and we're just taking the 0th and 1st order terms and ignoring the rest. Similar to how many things in physics are modelled as simple harmonic oscillator (2nd order), we're modelling changes to the metric linearly (1st order).
Yes there is an underlying limit operation. One can, and formally should, keep those small terms around during the manipulations. One just has to keep track of their magnitudes during all the steps (often it is not necessary to calculate them explicitly, just to obtain bounds for their magnitude). Then in the end one has a true equality and some "ugly" corrections. But for those we know how big they can be relative to the original perturbation (displacement). They should be orders of magnitude smaller, at which point we may just want to drop them.
Often some technical assumptions must be added to make it all rigorous.
You also have to remember that the small terms would be so physical small that in the case for GR, they might be smaller than Planck length meaning they "possibly" would be completely universally insignificant. So 0 is acceptable, that at least how I mentally justify it to myself.
On 11:10 we have derivatives with ~ acting on the metric h. In order to write them as ordinary derivatives I get an extra term of h_{αβ;κ} =h_{αβ,κ} +h_{αβ,γ} ξ^{γ}_{δ}(partial x^{δ})/(partial x ~^{κ}). I've used ; to denote derivative with~ and , to denote normal derivatives. Basically metric derivative wrt ~ = metric derivative wrt normal coordinates+ terms that involve ξ * the inverse Jacobian* the derivative of the metric wrt to normal coordinates. In order to make this term vanish you must assume that the derivative of h wrt normal coordinates is very small which is plausible for sure, but I'm not sure if we've assumed it. I tried to do this myself, got stuck there and then decided to watch your video, but I think you didn't address it. Is this an extra assumption we must take?
I have gone through your Tensors for Beginners and Tensor Calculus videos which are excellent. This series is also monumental. The explanations are clear and complete. If only some professors could do so good a job! Question regarding gravitational waves: You have demonstrated that the Einstein Field Equations reduce to a wave equation under the Lorenz Gauge transformation, so in a vacuum, where the energy-momentum tensor is zero, waves in space time are propagated the same as EM waves in space. In the case of EM waves, the wave is initiated by a boundary condition like an alternating current in a wire. In the case of two enormous masses (black holes) orbiting each other, what does the Energy Momentum Tensor look like? Are the terms sinusoidal functions of time?? Your insight would be appreciated.
I honestly don't know. I know it has something to do with a "quadrupole moment". But I haven't investigated it.
@@eigenchris I am going to dig into this and if I find anything, I will pass it along to you.
Incredible video!
If my tax is used to compensate eigenchris, I'll be proud of my country.
Hi Chris, Thanks from the bottom of my heart for the wonderful lectures! I am enjoying each bit. @14.42 we have a recipe for the displacement field as a solution of another wave equation. My questions are follows
1. How do we know the solution for the displacement field exists?
2. Even if we know the solution exists it is still very vague for me to be able to find the solution. Do you use a Green's operator?
3. What will be the initial conditions for such equation ?
I think this is the first time in your videos that you explicitly deal with a displacement as a coordinate transformation. I always understood the coordinate transformations you used to basically be linear maps.
And I actually made the assumption that tensors are only invariant under those, and not those that change the origin, but I see why that would be wrong.
But is there something else we have to pay attention to when we use something other than a linear map as a transformation? Or is that just trivial?
You can use any coordinate transformation you like. The basis vectors are just the tangent vectors along the coordinate curves, and tensors will transform according to the jacobian and inverse jacobian matrices. I think I already showed this in the 105 videos when we dealt with Rindler coordinates. Tensors live in the "tangent space" attached to a given point on a manifold. Changing coordinates of the manifold only changes the basis of the tangent space, it doesn't change any actual geometry.
@@eigenchris thanks for the clarification.
You are amazing ! Thank you so much for this
This was great! I have a question regarding the approximations made to get to this point. So first, you walked us through a derivation of the linearized Einstein equations, which is a perfectly sensible thing to do and very common. But I'm confused about the derivation of the Lorenz gauge in linearized gravity, since it seems like that's only an approximate gauge, at least as you walked through here, since you made assumptions about terms being small. This confused me since it seems like even in linearized gravity, the Lorenz gauge is an approximation, and possibly not satisfied exactly by any real transformation. Is that true?
Sorry I must have seen the notification for this comment, but forgot to reply. It's true that everything in linearized gravity is an approximation. The real Einstein Field Equations are non-linear. But the "linearized" approximation is considered "good enough" when we are dealing with spacetime curvature that's relatively weak (gravitational waves from distant sources are very weak). The non-linear effects are assumed to be small enough for gravitational waves that they can be largely ignored.
Is there a concrete simple example of a Lorenz Gauge frame? I understand that it's just a simple way of writing the field equations in a way that makes it apparent that it is a wave equation, but I would like to picture what kind frames satisfy the Lorenz Gauge.
In the next couple videos, we'll see a specific example of a Lorenz gauge called the Transverse-Traceless gauge. We'll use the metric equations that pop out of that to get formulas gravitational waves. I honestly haven't calculated specific Lorenz gauges except this one.
Thank you so much! I might've missed something -- how can we choose xi such that the partial derivative of h w.r.t. beta is the d'Alembert Operator (slide 31)?
Xi is an arbitrary displacement field. We're placing this constraint on it (making it less arbitrary) to get the type of transformation that we want.
Muchas Gracias 🙌🙌🙌!!! Excelente explicación 👌
Thank you Cris, for another great video and explanation! I have a question about the gauge invarience in GR. The gauge invarience in the EM filed results in charge conservation. What is the quantity conserved by the gauge invarience in ( curvature?) the Riemann Tensor?
I'm not sure I'm knowledgeable enough about gauge theories to answer this properly. I can maybe say a few things that you can use as a starting point to google more.
I know that in QFT, we normally view spacetime as having an "internal symmetry group" at each point (e.g. U(1) for QED). Taken together, spacetime (called the "base space") and the internal symmetry groups (called the "fibers") are a mathematical structure is called a "fiber bundle". A choice of gauge in in QFT is the same thing as making a choice of "connection" between the fibers (this choice of connection immediately leads to a choice of covariant derivative). In GR we don't really "choose a connection" or "covariant derivative" like this... there's only one connection/covariant derivative we every use, which is the Levi-Civita connection. So I'm not sure if the gauge invariance in GR is completely analogous to the gauge invariance in QFT, and I'm not sure if we end up with something analogous to a conserved charge. This is pretty out of my depth though, so I can't really say one way or the other.
@@eigenchris I think that's a good starting point! Thanks again.
my god, this was a good one!
I only understood about 10% of this, but even so I think I got the punch line.
By choosing the right gauge the equations simplify so it is much easier to understand, and although the answer is not perfect, it is close enough for government work.
That's the idea, although I'm sorry that you only understood 10%. Is there a main blocker to your understanding? Is it all the tensor algebra?
@@eigenchris It is mostly due to having a bad memory in that I watched many of your videos a while back but then promptly forget what I learned.
My other problem is while the notation you use is quite elegant, it is hard to keep track of what all those indices mean.
There may be a way to help fix that but would probably require some sort of tool that could link an index to a graphical (as opposed to symbolic) representation of it.
Thus say just in a x1, x2, x3 vector say one clicked on the 1 of the x1 it would draw a line from that to the X1 axis and if one clicked on the X of the X1 it would draw a line form that x to the vector X1 along the X1 axis sort of like a tool tip or like a here you are at the mall map.
Thus one could click on any symbol and have a line drawn from it to a diagram that shows what that symbol represents.
Unfortunately, as far as I can tell, no such tool exits, but it would be cool if it did.
Interestingly enough, I just made a RUclips video that sort of illustrates this basic idea, where those lines I mention above are what I call yarn that go with what I am calling bookmarks where the yarn is sort of like the yarn one sees in a detective show where they have a forensic board with pushpins and yarn to make connections.
Here is a link to my video that shows this (which albeit I sort of messy in that my handwriting is as bad as my memory.
ruclips.net/video/TsneUuzapIY/видео.html
And here is a link that shows a forensic board (or murder board) that illustrate what I mean by yarn.
Thus a bookmark is the virtual way of making connections like one does on a murder board, but instead of photos of suspects and the like, one could links indices to drawings using it.
ruclips.net/video/7IrBQFmAUi0/видео.html
With an aid such as that I might be able to connect the dots a bit better between the notation and what that notation denotes without having to rely so much on my (bad) memory.
Try doing exercices, there's only a limit of what we can learn just by listening to someone, no matter how well they explain.
I learnt it the hard way at university, by realising that I could ingest an immense amount of lecture, but that I would only begin to truly understand (and remember it) when doing the exercices (I have the chance to have university that put a big emphasize on exercices and I was surprised to see it's not so everywhere).
@@jonasdaverio9369 Agreed.
Quick question : can a gravitational wave go through a black hole or will the black hole "cut" the wave ? I mean, will there be a lens effect ?
That's an interesting question and I'm afraid I don't know the answer.
I dont know the answer either but if the waves could and there would be a lens effect, would it then not be possible to obtain information from inside the hole?
@@imaginingPhysics
Yes that's one of the many paradoxal consequences that crossed my mind, not sure what it would be. It also raises the question if even normal gravity (not wave) could be affected (even very slightly) by any body on its path.
@@eigenchris
Thanks for the answer, I know these are tricky questions and it's hard to answer these ;)
@@En_theo maybe you would find this interesting:
ruclips.net/video/cDQZXvplXKA/видео.html
Would it not be a mistake to suggest Inertial tensors are zero is other inertial planes exist
Any insight how we might interprete Lorenz gauge cond.? Its a "continuity equation", right. For the EM 4 potential it can be interpreted as choosing coordinates where the "scalar potential (the zeroth component) is conserved". This can be seen by integrating the divergence equation 21:55 and using divergence theorem.
So, could this metric Lorenz gauge be interpreted as choosing a frame where "the metric is conserved" in the sense that the contractions cannot just vanish or appear out of nowhere? Metric must propagate if it wants to change. This would make sense if we want to see Gwaves?
The "continuity equation" makes sense for vectors--it just means whatever goes in a box must leave. The continuity equation for the metric perturbation, however, I'm not sure how to make meaning of.
Can you explain what you mean when you say "contractions cannot just vanish or appear out of nowhere"?
@@eigenchris Ill try. Generally, if you move at x direction, g_xx tells you how much your x-coordinate changes. But your x-coordinate can also change if you move at y direction; g_xy gives gives you that rate. And g_xt tells you how your x-coordinate changes with time. Right? So we have "a set of meter sticks" g_xt, g_xx, g_xy, g_xz. Same goes for inverse metric. Now I think we can form a vector
(g^xt, g^xx, g^xy, g^xz).
It is not a 4 vector, but its not meant to be. We do want it to change, and we are looking for a frame where its divergence
g^xt,t + g^xx,x + g^xy,y + g^xz,z
is zero. And what does it mean when its divergence is zero? I think it means that if meter stick g^xt is stretching, some other meter stick must be contracting, and vice versa. It also means, I think, that local changes in metersticks cannot spring out of nowhere; there had to be fluctuations going on near by.
@@imaginingPhysics That's a very interesting interesting way of looking at it. And indeed, with a "plus"-polarized GR wave, any "extension" in the x-direction needs to be "stolen" from the y-direction, where a contraction happens. I can see that making sense. The only missing link for me is this sentence: "But your x-coordinate can also change if you move at y direction; g_xy gives gives you that rate". I normally think of g_xy as measuring the angle between the x and y basis vectors. But maybe the two concepts can be made to be the same?
@@eigenchris Yeah g_uv as an angle is more accurate interpretation. My language was sloppy and incorrect as I tried to explain the idea: Of course if one moves at y direction then by definition only the y-coordinate changes.
Still, if x and y axes are not orthogonal, then there clearly is a sense in which one seems to going in x-direction also (draw a picture). And this is what the metric component g_xy measures (because if represents the angle).
At the moment I don't find better words for it. Covectors ("stacks") might be useful too.
at 20:05, are the electric and magnetic vector components supposed to be covariant?
I'm not sure if I was worried about covariance/contravariance when I wrote those. The formulas at 20:05 are just the standard formulas from a non-relativistic E&M class. If you're going to do E&M in a way that's Lorentz-invariant, E and B fields are the wrong way to think of things. You need to think in terms of the Faraday tensor at 20:45, which is a twice-covariant tensor.
Thank you Sir ❤️❤️❤️
(Sorry, tried Mathtype paste here -no joy) ...have you ever thought of (x alpha) and (x alpha prime) as yr x co ords instead of (x) & (x tilde)...it greatly simplifies comma=derivative notation
totally avoiding ambiguity
I started using x-tilde in all my videos because i found it easier to read than the tiny prime symbol. In this video it makes things abit worse, but I stuck with it for consistency.
@@eigenchris Totally understandable! BTW just an acknowledgement of the great work you've done. Esp yr analogies on covectors (sthing I always struggled with)& of course the SR & GR work. Have you ever read Quanta magazine? A little above New Scientist but good on Quantum Gravity & related stuff. Another interest is Unified Field Theory (Mostly above my head!)but there are 2 excellent 100 page papers on its history up to I think the 1970s.(online)...can dig them out if you're interested.
Brilliant 👏 👏 👏
When one contracts the wave equation for h_bar with the Minkowski metric, one gets an equation that says that the d'Alembertian of the trace of h is zero.
Can I plug it in in the first wave equation to simplify it to a second wave equation which simply says that the d'Alembertian of the tensor h is zero (as opposed to the full tensor h_bar with the sum of the trace)?
In the next video I plan on doing something along these lines... although I haven't fully thought it through yet. I'm not sure if you get that for free in the Lorenz Gauge or if you need to pick a special coordinate system.
@@eigenchris I think you get that for free in the Lorenz gauge and another way to see that I think is to consider the Einstein Field Equations in the vacuum, and they reduce to the Ricci tensor being 0 (the space is Ricci flat), and one can do the same calculations done in your vids, imposing the Lorenz gauge afterwards to get the wave equation simply for the disturbance in the metric h.
Or am I making this up? :P
@@tommasoantonelli7176 Yes, that does seem to make sense. I think I was getting confused because in the next video, I plan to set the trace of "h" itself to zero. But in an arbitrary Lorenz gauge, even though "trace of h" isn't zero, the "d'Alembertian of the trace of h" does to go to zero based on the reasoning you gave.
So we chose a coordinate system that obeys a wave equation to get a metric that obeys a wave equation. How can we be sure that the waves this predicts are really a feature of the metric and not a coordinate artifact?
Isn't that why Einstein changed his mind about gravitational waves like 15 times?
In 109e we'll look at how the proper distance changes between parallel geodesics, and we'll see it vibrates along with the wave. This is a result that doesn't depend on coordinates, and it basically means free particles will follow the animations I show in this video. It's a good question though.
@@eigenchris thanks, looking forward to it.
Will you also talk about what the fact that it moves particles means for the energy of the wave and what this means for gravitational energy as opposed to stress energy? Or is that outside of the mathematical scope of this series?
I've been wondering if there's a way to get an energy tensor that includes gravitational energy.
@@narfwhals7843 I don't really understand the mathematics of how gravitational waves carry energy yet, and I wasn't planning on making a video on it on my first run. After all the main videos are done I might make a GR video once in a while to answer questions like this.
Let's go!!!
Fabri parot cavity
Can you upload this ppt to Github?
I can get started on uploading some of the relativity notes there.
Voice and pronunciation of greek letters are horrible! I could not stand them.
Sorry, I'm just pronouncing them the way they are normally pronounced at Canadian schools.
@@eigenchris Actually, I have to be sorry for my comment. It's easy watching from the outside to criticize glitches in a video or explanation. Were we should focus is on the content and usefulness of the explanation or information content. My apologies.
@@ascaniosobrero It's okay. I hope you get something out of the videos.
Awesome maestro 👏🏻