The Surprisingly Subtle Limits of General Relativity
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- Опубликовано: 26 сен 2024
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Einstein's theory of gravity---general relativity---was the last great pillar of pre-quantum physics. Gravity, he says, results from the curvature of spacetime in the presence of massive objects like stars. Then lighter particles moving nearby just travel along the straightest possible routes that they can through this curved geometry, called geodesics. And yet over 200 years earlier Newton wrote down a vastly simpler force law for gravity and used it to derive the orbits of the planets in our solar system to remarkable accuracy. So how is it that these two radically different descriptions of gravity fit together? And how does Einstein's theory go beyond Newton's, to predict the existence of black holes? (Like the Sagittarius A* supermassive black hole at the center of our galaxy that the Event Horizon Telescope collaboration released the first ever picture of just last week!) We'll explore it all in this video. We'll learn about geodesics, gravitational time dilation, the Schwarzschild solution of Einstein's equations, and the black hole geometries that emerge when you squish a star into a ball smaller than its Schwarzschild radius. You'll come away with an appreciation for how Einstein's theory can capture the same physics as Newton's inverse square law in the limit when gravity is weak, and at the same time describe black holes in the opposite limit when gravity is strong.
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I’m Dr. Elliot Schneider. I love physics, and I want to help others learn (and learn to love) physics, too. Whether you’re a beginner just starting out with your physics studies, a more advanced student, or a lifelong learner, I hope you’ll find resources here that enable you to deepen your understanding of the laws of nature. For more cool physics stuff, visit me at www.physicswit....
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Love the vids, can you please do a playlist on tensors or just explaining tensors and basic tensor analysis?
@Michael Bishop qqqqqqqq
000⁰000
😊
O
One thing I really love about this channel is the fact that it’s unafraid to get involved with the Math whilst explaining the deep concepts underlying the Universe.
It’s so easy for an educational channel on Physics to go purely for the concepts, without explaining how or why they came to be, so that the viewer walks away thinking they learned something, when in actuality the content went in one ear and out the other.
On the other hand, it’s also a pitfall for someone knowledgeable in the subject to delve right into Maths without explaining any of the underlying intuition or concepts. Useful for students trying to solve university problems, but not for an educated public seeking to further their understanding.
This channel has the balance just right! Great job, Elliot!
Thanks LM!
You've produced a very high-quality video. This is one of the best explanations of gravity I've seen! I've watched it several times now.
I appreciate it RiteMo!
I'm a 2nd year Physics student. Ive just discovered your channel and I'm watching every video you posted, one by one. It's pure gold! Thank you so much and please keep up the amazing work!!! 💚
Love the video so much!! Would you be interested in going through the math that shows how GR explains Mercury’s precession? I’ve been wondering about it forever!
Thanks Maxx! That's on my list of potential topics, maybe someday
That is pretty simple, the Schwarzschild metric is in spherical coordinates (radial coordinate r, polar coordinate theta and azimuthal coordinate phi), you use one of properties in this metric: conservation of energy and angular momentum, to find a differential equation dphi/dr, and you solve the equation, from the solution you will find the formula for procession (instead of closed elliptical orbit, it is open), substituting numerical values into the result, you will get Mercury's precession
You have to be a special kind of gullible fool to believe in EInstein's nonsense theories. Don't you realize that its all a fraud? Physics and cosmology today are driven by fraud and deceit. Also Medicine and a bunch of other of the sciences, which are now all pseudoscience. A great tool of propaganda.
I think Eigenchris made a video on it in his general relativity playlist
Well constructed. For ordinary folk, with limited math skills, it takes several viewings to get the flow. I'll persevere.
I'm bewildered by how much physics terminology and notations have changed from when I studied Special Relativity in the 1990s! 😵
I was taught that objects _gain_ mass when they move relative to you. The _total mass_ is the sum of the _rest mass_ and _relativistic mass._
Today we don't use those terms anymore. Rest mass is just mass, and what the object has gained is simply energy according to Einstein's less famous but more complete equation E² = m²c⁴ + p²c². (Back then, we used to write m₀ instead of m in that equation.)
There are many more examples like that.
But the biggest change I see is in how the spacetime 4-vector is defined nowadays: [ct, x, y, z]. Back then, it was [x, y, z, w], where _w = ict, i_ being the imaginary number √-1. The final results were the same, but using _ict_ (and identity matrix) instead of _ct_ (and Minkowski matrix) made the equations symmetric, uniform, and more intuitive. Time dilation, length contraction etc. could be derived as a straightforward change of coordinates-specifically, a rotation-in 4-D spacetime.
I'm okay with ditching relativistic mass. But there had better be some good reasons why we don't use _ict_ anymore! 😡
To be clear, I'm an engineer, not a physics major. But I did study (some) Special Relativity and Quantum Mechanics in college (but not General Relativity). The courses were proper Physics 101 courses offered to all students, and taught by professors from the physics department. So I believe it was the real deal, not a dumbed-down version taught to non-physics students!
Yes relativistic mass is an old idea that isn't used anymore!
The i is used to go back and forth between Lorentzian geometries with the negative time term and Euclidean geometries. So from four dimensional Euclidean space with metric ds^2 = c^2 dT^2 + dx^2 + dy^2 + dz^2 we can transform to Minkowski spacetime by continuing T = i t. It's known as a Wick rotation.
This is such a well made video, having an extremely precise scope and amazing narration. I don't know how deep you can go into GR with this format, but I sure hope you do.
I saw the thumbnail, thought it was some BS popular science BS handwaving explanation, clicked, and was in awe. Thank you for satisfying the minority who loves seeing the hairy math. Subscribing isn't enough after seeing the rest of your content.
Glad you liked it Jae!
About 04:20: _The result is an ellipse with the Sun in one of the focus points._
To sufficient accuracy, yes. Actually it's the common center of mass but since Sun outweighs Earth by a factor of over 3×10⁵ and all other masses of the solar system still by about 500, it's fine.
Also, the statement about 05:15 is true if m
I was with you right until you started talking.
Such an excellent video! BTW The Schwartzchild radius is computed with the assumption that all the mass is concentrated at a single point in the center. So the real reason the equation doesn't work inside the star is because of mass, not radiation. In fact, inside a (uniform and spherical) massive object, the inverse square law also doesn't directly work and the force is proportional to the radius (like in harmonic oscillators). That's because the mass from outside of the radius is pulling from the opposite direction and cancelling the gravitational force.
The way you have broken down the components of GR makes it easy for a person like me who is not a physics major to understand. Thank you Elliot, Shoutout from South Africa
Glad you liked it Sphakamiso!
! Thank you very much for the open access to the notes! Finally, I don’t have to take notes on everything myself, so as not to get confused later, and it didn’t seem that I could forget something)
Amazing man amazing😭😭😭😭I can't believe that these things are actually getting into my head, please keep making such free contents 🙏🙏
I've decided that I am going to learn everything possible about Einstein's theory before I die someday. If I have to master tensor analysis and differential geometry, so be it.
Read his actual publications translated into English. Despite the math being insane, he was actually a really good writer and thought experiments don't require a physics degree.
I have just been reading a book titled 'Black Holes' by Brian Cox & Jeff Forshaw which includes concepts such as flat v warped space time, Schwarzschild solution, etc. As a maths graduate I wanted to understand the maths behind this. Your video has done this admirably for me in the space of 32 mins! Bravo! I look forward to your video on black holes.
Nice one, Elliot! I think that's quite a good explanation! Keep up the good work man.
Thanks Aadhi!
Thank you. Something I've been wondering is how Einstein calculated the precession of Mercury without knowledge about the Schwarzschild metric. I figure it might have been by adding higher order epsilon terms using the value of epsilon computed in the Newtonian limit.
I'm not familiar enough with the history to answer, but he may have only needed conservation of energy and angular momentum, and the weak-field limit of his equations. You might try looking at Pais's biography of Einstein for some of the history
This is one of the best videos on the basics of General Relativity. Such wonderfully crafted and framed with so much clarity and care at different junctures!
Very useful video. Thanks from the heart for putting this up.
You explained masterfully on how to get Newton's second law from GR in the appropriate limit
Thanks Cesar!
Your channel is awesome, the way you explain and narrate physics is precise also you teach deeply with a lot of details, I completely understand the special subject you teach whether basic or advanced, love from Iran 🇮🇷 ❤💫🥰😍😘
Thank you Zahra!
Oh, I've been waiting for a GR video!
Hope you like it Roxas!
After AP physics in college, this is a good topic to learn further about physics and the universe.
Great video - love seeing the equations of GR reduce to Newtons laws
Excellent video, very easily explained such a complex subject.
Excellent video. About planet orbits: the orbit of planet Mercury around the Sun is not exactly described by Newton’s law, while it is by Einstein’s equations.
I appreciate your hard work sir.... These videos are great and highly informative....
In GR, the (timelike) geodesics followed by test particles are the longest paths through spacetime, not the shortest.
Nicely developed explanation showing that Einstein's equations reduce to Newton's equations in the appropriate limits and all of this is done without any handwaving glossing over of the mathematics. Well done and thank you!
I think that Newton had the idea that gravity bends light through a simple thought experiment: Bob, standing next to a wall of a free falling elevator shoots particles of lights towards the opposite wall. While Bob determines that the particles moved in a straight line, Alice on the ground, does not see the same. Instead, she sees the particles moving on a curved trajectory! Therefore, they conclude that light moves on straight lines on frames free from the action of forces, but not when moving in a gravitational field. Therefore, gravity should bend light! Furthermore,they conclude that the greater the gravity, the more curved will be the trajectory.
Finally some GR! Hope you go more into the details in the future!
Great explanation especially in geodesic equation derivation..... 👍👍... I saw similiar video years ago but not so Details explained... Thanks a lot Dr Schneider for your huge effort to make such great Videos and upload in RUclips so everyone could understand Physics.... 👍👍🙏
Much appreciated Steffen!
I don't understand, BUT oddly, i found that very interesting
Touchy topic, thank for clear explanations. I guess many people wish they had you as a teacher.
Great video. However, some statements prompt me to some remarks (you will do it themselves later on):
07:05 f: _It follows the shortest and straightest paths it can._
Since we are talking about spaceTIME, we aren't so much talking about its (spatial) path but rather about its _world line_ which contains even more information: May a path be straight, the particle still could be linearly accelerating (which includes deceleration since in physics, acceleration means _any_ alteration of velocity) for some reason but its world line would stop being straight.
BTW, due to MINKOWSKI's metric, the straightest chunk of a world line between two points (events in that case) is the longest rather than the shortest.
07:25 f: A circular or elliptical orbit is _not_ the straightest spatial path something can travel. This honour is due to the hyperbolic path of light passing Sun. However, the _world line_ of the particle or planet, given some initial position and direction (which means velocity of the body), is the straightest possible.
Best posbl explanation of GR
So I need to come up with a new theory of gravity that also reduces back to Einstein's theory as well. Alright, I'll take on that challenge.
One surprising thing I learned during my studies of general relativity is the "Newton-Cartan Theory", which lets you describe Newtonian gravity using curved spacetime instead of force vectors (you end up with a Ricci tensor that is zero except for the 00/tt component, just as mass density is the 00/tt component of the energy-momentum tensor). It makes the jump to General Relativity seem much less radical.
how are you not verified yet lmao
I love your relativity and tensor analysis series. I can sit and listen to it for ages.
I've even changed my American pronouncing of "z" to "zed."
It's fascinating that it's the warping in time that's responsible for the Newtonian limit!
what is the reasoning behind it being okay to flip the plus and minus signs at two places like this?
General Relativity describes HOW matter and energy curve the spacetime. But is there an explanation of WHY matter and energy curve the spacetime?
Thank you for the video.
Thank you Elliott, very good episode!
I have been wondering, if gravity is a property of spacetime and not of the object traveling, the straight path should be the same for the earth traveling around the sun and a light ray traveling in the same direction,isn't it? So why does the light keep traveling straight, while the earth path is bent?
Light bends too! That's called gravitational lensing
it's the same reason why a light ray can travel straight up off of earth's surface, but if you throw a ball straight up it will fall down.
You have to consider its path through spacetime, not simply space. Earth and the light ray are not traveling in the same 'direction' through spacetime; their different velocities correspond to different 'angles' of travel. (Picturing a Minkowski diagram might make it clearer. Or possibly not; explaining through text alone is challenging.)
I like this question.
The question makes it obvious that following a geodisic in space time isn't a one size fits all. The geodisics for different speeds differ - my brain is beginning to hurt. Doing the calculations and intuitively understanding the results aren't the same.
The confusion starts, I think, with the word "straight", I think the curved space time approach says there is no such thing. There is a more general concept of geodisic which is a minimal route.
Let's suppose that the Sun could spontaneously disappear and that thought experiment undermines Newton's law of gravity. But, it also undermines Newton's laws of motion. In particular, the conservation of momentum. In a reference frame in which the Sun is moving, it cannot spontaneously disappear without violating conservation of momentum.
Likewise, if the Sun spontaneously disappeared, then the Einstein field equations would be violated, as they imply local energy conservation.
In summary, no physical theory can support the spontaneous disappearance of the Sun, so it can't be a valid thought experiment!
One thing i haven’t been able to figure out is, as the explanation of objects X as the result of geometry and not a force, is this more a reflection of the math we use to describe the universe, or a transparent view of our physical reality
General Relativity is illogical nonsense.
The way physicists talk about this is very misleading, but generally for non-weak fields, and non-asymptotically flat spacetimes you don't get to split the metric like here, and g_00 can't be interpreted as a potential
Why does an object thrown from different heights hit the ground at different speeds? If there is no acceleration in the geodesic
«Why does an object thrown from different heights hit the ground at different speeds? If there is no acceleration in the geodesic»
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A very smart question, though a waste of time. Anyway nice to see genuine questions by genuini thinking, in our Brave World Of Parrots and their clickbait maitre-a-depenser.
Really nice video! I'm so happy I just started learning about special relativity...and I can't wait to delve more into its details!!! Have a nice day :)
Geodesic is only a model of gravity. Gravity is a force and space does not curve.
Well done sir.
I love this channel man
4:50 got it! FU=Gravity😅❤
This be brilliant
4:11 if F is negative, shouldn't U be positive?
Using F = -dU/dr, there will still be a minus sign after taking the derivative :) The potential is lowest at the origin r=0, creeping up to zero as r --> infinity. The force on m points follows the gradient to lower U, toward r = 0.
Amazing video!
11:50
It would be much easier and more logical to use the other sign of the metric, where cdt is the positive one and dx, dy, and dz are all negative. Then all distances in spacetime are positive already and there's no need to negate it before taking the square root and integrating. Even though it really doesn't matter too much as long as you're consistent, I find it makes the math less confusing.
Sounds like a good suggestion. Without swapping the signs back to the original magnitudes, would it cause problems with interval lengths and vectors being "time-like" versus "space-like?"
Mostly plusses is the relativist's convention - mostly minusses, the particle physicists'; personally, I prefer the first
@@thstroyur I prefer practicality and simplicity over looking at a bunch of minus signs.
Photons are said to have mass...is there any force at the point a transmitting electronic circuit emits an electromagnetic wave ?
Can you make a video on explaining why we need the idea of manifold and what is a manifold in general relativity?
Maybe! A manifold is the mathematical construction of the spacetimes we were discussing here
Woww dude it soo good dammnn. i love you and your contents 🔥🔥
Thanks Tanvir!
Will you make a video on relativistic QM and the dirac and klein gordon equation and stuff?
ruclips.net/video/13hCkUiu_mI/видео.html
please share the perfect sequence .....from where to start general relativity upto end???
Wonderful!
More videos please!!!!
Very soon! Keep an eye out
Amazing
Thanks Wasi!
when is the next video coming?
Once I've finished the course I've been creating on Lagrangian mechanics. In the meantime make sure you're signed up for my newsletter for weekly little lessons! www.physicswithelliot.com/sign-up
A masterpiece
Thanks Prof!
Great video! How about the vorticity equation?
Hey Elliot!
I'm a high school student and am very interested in physics. Would you please recommend some books to read to clear basics as well as grow deeper understanding in physics?
Looking forward to hearing from you.
Have a good day
That's great Matilda! Make sure you check out my "help room" playlist videos, which are aimed to be more introductory. Picking a book would depend on how much math and physics background you have so far.
30:00
Great video as always... I'll never get as good at physics as you... lol
Thanks CG!
Flipping the sign of the Minkowski metric hides the fact that ds is imaginary for a stationary object.
What else happens if you stick with this metric and not switch the sign?
Proper time would be imaginary.
You would be perpetually late or early for your appointments.
Excellent video Elliott. I have a question pertaining to gravity. How can gravity be considered not a force? As an excample, if one passes the event horizon into a black hole, is not this attraction a force?
I was actually thinking of talking more about that in my next email newsletter. So make sure you're signed up!
@@PhysicswithElliot I did thanks. Today's scientific community in this field seem to call it an emergent force one that bends space time and not a direct force. Seems odd to me. A direct stationary retractive force that starts from a beginning before the bending of spacetime is still a direct force and I would conclude this in a deceptive black hole of immense direct retractive force.
Why does the Minkowski metric take the negative of the time term and positive of the spacial displacement terms? Wouldn't it make more sense to keep the time term positive and subtract the spacial displacement terms?
Since it's a distance wouldn't it essentially be the same thing too? Should we be taking an absolute value?
You can write it with either overall sign---it's a matter of convention. But the way I've written it, it gives back the usual spatial metric +(dx^2 + dy^2 + dz^2) when you look at a fixed time slice. With the other convention you'll get minus that
Why is the combination of \epsilon_{\mu,
ho} with \frac{d^2 X^
ho}{dt^2} second order in epsilon?
Because the acceleration itself is already first order in epsilon
Did I miss the part about what happens to the Earth if the Sun disappears under Einstein theory? I’m unclear as to why Newton’s take would be so obviously wrong, so was waiting for it and it didn’t seem to come.
The effect would propagate outward through the gravitational field like a wave pulse, whereas Newton's law would erroneously say the field changes everywhere instantaneously
The very geometry? There is no “geometry,” some magical fundamental property of space. Space is space and gravity doesn’t bend it. What bends is dimensions, our measurements.
«The very geometry? There is no “geometry,” some magical fundamental property of space. Space is space and gravity doesn’t bend it. What bends is dimensions, our measurements.»
--
Nice, though "model's measures" - that there cannot be a measuring instrument long enough to measure objects inside those abstruse concepts inside abstruse models. As Tesla said, space cannot have whatever property. I was thinking the same before reading this one of Tesla.
Well, that's probably besides the point, but finding solutions of Einstein's equations is actually pretty easy. You just choose a metric, compute the Einstein tensor, and then set the stress-energy tensor equal to it.
What's hard is to find a solution that describes a relevant physical system.
If the earth follows a geodesic path( this geodesic has popped up from a non accelerated body hypothesis)...but we know the earth accelerates near the sun thus how can it follows a geodesic?....
It's the coordinate frame we're using. It's similar to how, when we're in a car going around a curve, it seems like there's a force pushing things outward, when in reality we're viewing things from an accelerated frame of reference. Also, we need to consider Earth's path in four-dimensional spacetime and not simply three-dimensional space.
@@tomkerruish2982 The parroting won't change the fact that models are not reality, but just representations. The trick of posing as understanding what's going in four-dimensional model (or pseudo-reality) is going to add more cognitive confusion to you than on those who read (that people are smart enough to ignore what is not pertinent). I have some genuine questions for the geodesic space-time adepts.
@@voltydequa845 Okay, I'll bite. What are your genuine questions?
@@tomkerruish2982 «Okay, I'll bite. What are your genuine questions?»
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I have cold. Let's start with the tide.
@voltydequa845 You mean the effects of a nonuniform gravitational field?
Newton: A nonuniform gravitational field results in different parts of an extended body experiencing different accelerations, thus moving (or attempting to move) in different ways.
Einstein: Nonuniform curvature results in nearby geodesics converging or diverging, thus causing parts of extended bodies to move (or attempt to move) in different ways.
Edit: I hope your cold gets better.
Superlative pedagogy
Never been so early
Gravity: Imagine a sheet called space moving through a 4th dimension where the 4th dimension was called time. The fabric’s direction of travel is from a place we call the past to a place we call the future. The surface of the sheet is what we will call the present. This 4th dimension is not the same as our ordinary experience of time but is related to it. If you put a lead ball on the moving fabric the ball would resist the movement so as to cause the fabric to deform. The ball and the fabric pushing the ball would lag behind the rest of the fabric. According to our definitions, the ball could be said to be in the past. If another ball were put on the fabric it would deform the fabric too. If the two balls were close together they would approach. The balls would seem to be attracted to one another but in reality, they would be both moving toward the same place.
«...but in reality, they would be both moving toward the same place»
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But in reality your is just re-wording of the reality. Anyway two attracted magnets are moving toward the same place. In reality they are moving toward the same place because attracted. It is depressing to see how we lose cognitive capacity by means of all this abstruse re-wording behind abstruse models that present themselves as "reality". We perceive the reality by means of descriptions, with all their limits.
@@voltydequa845 You are wrong.. You took ", they would be both moving toward the same place." out of context because of your own ridged thinking. They are not attracted to each other. they are not moving toward each other. A point in space in moving toward both of them. ruclips.net/video/XRr1kaXKBsU/видео.html
So...if clocks don't agree due to their relative motion, are we travelling into to the future, via our motion, relative to a 'stationary observer? If so, how can we communicate, I'm unaware of any mechanism that would allow me to have a 'direct' conversation with my future wife, since we don't occupy the same 'instant' of time. confused!
Hm? Signals still travel locally through spacetime. Two paths through spacetime may both start at point A and both end at point B, while the length of those paths may be shorter for one than for the other.
For signals sent between two people, just consider the paths the signals take as well as the paths of the people.
Einstein has few famous works
It didn't change the universe one iota.
elliot funny name 🤣🤣🤣
I used to believe all the time dilation nonsense until I grew up.
Kepler’s 3 “laws “ all force free and mass free. Neither Newton, LeSage, Mach, Einstein correct on gravity. “The Final Theory: Rethinking Our Scientific Legacy “, Mark McCutcheon.
bro if you watch this while on mushrooms it helps u visualise what this is all about wowww
Something is wrong there . Time will bring the truth . This gonna be unbelievable ,and then the theories of everything will be reality ,.........
Both Einstein and Newton have never explained the CAUSE of gravity. All they did is describe the EFFECTS of gravity mathematically. I only know two people, besides me, who know the difference between a description and an explanation. Mathemagix is a language of description.
FIRST
F1RST!!1!!
And all of it is a humongous error. 🥴
Einstein General Relativity theory is wrong.
No, it has been experimently verified and it is one of foundation of moderm physics
Not only that, faith in it is a religion.
@@romierclark7886 «No, it has been experimently verified and it is one of foundation of moderm physics»
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Verified what? By which instruments? Modern physics that does what (with what results)?
@@Chicken_Little_Syndrome «Not only that, faith in it is a religion.»
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Religious people have more flexible minds. Instead this kind of faith has to do with parrotism.
All explanations of gravity are wrong so it doesn't matter what they said. Black holes don't exist either, they aren't doing very well are they?
and why don't black holes exist?
@@ahusky4498 because there no such thing as a pull force. Gravity isn't a pull force... it flows inwards. So a black hole is most likely a negative mass hole.
@@pinchopaxtonsgreatestminds9591 I thought you said they don't exists. Now they're negative mass holes? I'm beginning to question that PhD in physics you have.
@@ritemolawbks8012 If they are negative mass holes then mass doesn't attract mass. Negative mass is the area of least resistance for mass. It makes the formulas wrong. The formulas miss the conversion of gravity into mass. It means that Earth is more like a sponge for gravity, and removes the bending of spacetime, It makes time the area of least resistance for gravity towards holes in mass. Then that eliminates the Big Bang, and makes a red shift the result of gravity reducing in size to escape from mass.
So basically everything is wrong if black holes are negative mass holes. The formulas would be a result of imagination, and not factual... yet still appear to work.
I was just wondering what kind of numbers appear in the g matrix that replaces the eta matrix in GR. Is it still diagonal? And how much does it differ from eta in limiting cases like black holes?
The Schwarzschild metric from the second half of the video is one example
It's not necessarily diagonal, but it's always symmetric about the main diagonal. You can always change your coordinate system to make it diagonal, however, in particular by choosing axes at right angles to each other.
Amazing
Yayy! We touched on this in school! Black hole hype
Yes!
Newton's equations of motion has one great advantage.....it is considered more intuitive a great starting point and limit point for theories.
Great video, everything flows easily from the Lagrangian formalism, guess at a Lagrangian and the formalism of the Lagrangian least action principle provides the equations of motion.
One big difference is that Newton's equations of motion aren't Lorentz invariant while Einstein's Equations are, or under typical circumstances they are, (experts can almost always find an exception)
If we could meet yoda ans us petting the dinosaurs... we could revive death.
Why no quaternions in these equations? Thank you!
That is so cool. I was trying to understand GR, and that was my first question, how it goes to Newtonian gravity. Regret that haven't seen it before
Thank you for the video.
Glad you liked it John!