The Spacetime Metric
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- Опубликовано: 7 авг 2024
- In the (epic?) conclusion to our metric tensor trilogy, we dive a little deeper into the origins of the metric tensor in an attempt to bridge the conceptual gulf that lies between our ordinary, intuitive understanding of geometry and that abstracter comprehension of it required to master General Relativity's four-dimensional spacetime. Can we pave ourselves a path to clearer understanding in twenty minutes? Let us know in the comments.
Contents:
00:00 - Intro / Describing Distance
01:56 - Maps and Metrics
04:28 - Intro to Spacetime Cartography
09:07 - Formulating the Metric
11:57 - Interpreting the Metric
13:57 - An Analogous Explication of Spacetime
16:11 - The Spacetime Metric
20:29 - Conclusions 
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Far and away the best explanation of the spacetime metric tensor I’ve seen, over 40 years of doing this
Exceptional work. Definitely one of the best on youtube.
Incredible video! Really gives me a precise intuitive roadmap should I want to deeply learn the mathematics of GR. I’ve been searching for something like this for awhile and seems like a first of its kind.
eigenchris has exactly what you need.
Please keep doing what you are doing.
It is really wonderful✨
Dear Dyalect. You're work is one of the best on RUclips and with this great quality it will grow very big without any doubt. Bravo 👏
Subscribed. My favorite moment was when you artfully explained the geometry of the space time map …” is rather a map of world events which are merely ordered in space and time .” 🤯🤗Thank you.
Thanks for subscribing!
Man, this should be shown in IMAX, the best sci fi movie ever made!
Aaand.. I'm going to watch it again.
Also I hope there's a sequel, with quaternions and stuff. Maybe even a trilogy!
Thank you! Next one in the series will be on the Schwarzschild metric!
@@dialectphilosophy Oh my goodness! Yesss!
Great series especially for someone like me who doesnt have all the underlying mathematics down to understand something like General relativity
Very nice work on the visuals and modelling!
Great use of maps and globes to keep it grounded in something familiar. Cool to see how Einstein keeps company with people like Ptolemy and William Clark.
Maps and globes are pretty abstract things though
I must ECHO the sentiments of all others - phenomenal presentation both the gestalt and production! Bravo!
Most definitely! Superb!
This is the best explanation I've seen for the Metric Tensor. It acts as a translator that translates the language of the terms in the metric tensor that are in a mathematical language I previously didn't understand into a language I do understand.
I've been teaching and studying metric spaces for over 45 years, and I have to say, this video, more than any other RUclips video on this subject, is a video on this subject. You explained the subject in English with visual aids. Thank you very much.
Thank you, that's very kind praise!
Simply a great explanation for a very difficult topic.
Appreciating my first "ohhh...I get it" event at 3:18 -- nice lead-in to that.
Thank you! Sometimes its funny that something so complicated seeming, like a metric tensor, can really be reduced to basic labeling of distances along every interval of your map
This is the most incredible didactic effort ive ever seen. You guys have to be awarded for this.
Totally brilliant. I'm eagerly awaiting the next video in the series. But I can see you put a lot of work into these videos and it takes a long time to make them.
One tiny improvement, the 'fun aside' that popped up was hard to read. Please make any fun asides in future videos easier to read. Of simply put them in the description.
Excellent work , Keep posting
That was absolutely brilliant. Thank you so much!!!
That was a superb video, it gives a good intuitive understanding of what the space-time metric is all about.
An understanding that you can touch, taste, smell, and feel!
Truly amazing, well done! Your content, but also the slow presentation with perfect pauses in-between ideas, along with the visuals, are top notch! 💯🙏
Crystal, elegant, and enlightening! Thank you for educating and delighting so many with clear devotion to your craft here. So clear!! Also clearly so much precise and careful work, but so worth it for such a product we all truly appreciate in the end. Hoping you do many, many, many more! Amazing! 💗
Gosh, thank you for watching! ☺ Not everyone finds the spacetime metric as interesting as we do lol, but guess it's all about trying to share what you love with others. Your videos are definitely exemplary in that regard.
Great Illustration.
Great job ..keep up the great work and release more videos more frequently 👍
WOW! That was a great journey. Thank you for explaining that so well...
Thanks for watching! More to come down the line...
I watch so much universe and space / time stuff, and so rarely do I actually feel educated. This was amazing and gave me so much context to integrate into everything I had previously listened to. Instant subscribe.
Thank you, really glad it helped!
@@dialectphilosophy No thank you, and keep it up, may take a while for the RUclips algo to be friendly to you, but amazing content like this will float to the top without a doubt.
Just subbed, So much hard work went into these videos, keep growing, love from India.
Always waiting for new India. Thanks for giving good insights.
Love your content and the quality it is presented in! Can't wait to see new stuff coming :)
Waiting for the next videos with the complicated stuff made simple!
Fantastic video animations ! The intro I have seen yet. Would LOVE to see similar treatment of the Riemann tensor.
Thank you! Riemann tensor is a little ways down the line still…
The most intuition-compatible essay about this very important and complex issue I ever found ! Thanks a lot !
To me one didactic problem remains, in fact AFAIU the one that imposed the conflict in the highly controversial video on misunderstandings among multiple contributions on gravity that can be found: What exactly is meant by "locality" or specifically "locally flat". (This potential ambiguity lurks in some corners of this video, too, as Narfwhales explained me, personally.)
The rather easy to see conflict is that the curvature of a sphere is the same everywhere (unequal zero i.e. the 2nd derivative never vanishes), while the (Euclidean) metric of the sphere needs to be considered "locally flat" (i.e. approximates that of the tangent plane, as you always can find coordinates such as the first derivative at that point vanishes). In curved Minkowsky space in GRT this results in the inability to find s shortest curve between two events and hence (other than in SRT) there is no well defined spacetime interval number, while the spacetime length of any curve is defined an independent of the observer. (BTW.: in SRT the spacetime interval is not calculated really differently than the length of a spacetime curve in ART, the algorithm just can be simplified a lot.)
-Michael
Thank you for watching, glad you enjoyed it! But no, there's not any conflict with non-zero curvature vs. local flatness. Curvature is closely related to how the orientation of a tangent plane changes from point to point, so although tangent planes are always flat (this is what is meant by local flatness, and there are no ambiguities lurking there, at least mathematically speaking), how they are orientated will always change from point to point on a curved surface (no two tangent planes on a sphere for instance are directed the same way). If you don't prefer viewing manifolds as embedded in external spaces, you can adopt a parameterization perspective.
Be wary of trusting commenters who like to act as though they like they are expertly informed on such subjects, as there are quite a few lurking on this channel!
@@dialectphilosophy Of course tangent spaces (planes) are flat, and of course they might or might not be tilted against each others near a certain point of a manifold.
But that does not explain / define the term "locally flat".
I *suppose* a sphere is considered "locally flat" everywhere, even though the tangent planes nearby a given point always are not parallel.
I *suppose* a cube is not considered "locally flat" at the edges, as there is no unique limit for the tangent space for any sequence of points convergent to a point in an edge. (I never saw this discussed anywhere.)
And as long as the definition of locally flat is not given, I think implicitly using it might produce some ambiguity.
@@michaelschnell5633 The idea of "locally flat" is well-defined - loosely speaking, if you can approximate a portion of a curve with a line (or manifold with a tangent surface etc.) then it is locally flat. The corners of a cube do not "look" flatter the more and more you zoom in, so they are not locally flat. (Part of the definition locally flat includes continuous derivates.) We'd refer you to a calculus textbook (or you can probably find it on wikipedia) if you want the precisely formulated definition.
@@narfwhals7843 The comment wasn't in reference to anyone in particular, and we apologize if we gave you offense. We welcome all thoughts and opinions on this channel. We just noticed a sudden influx of commenters after publishing our gravity video who, despite lecturing to other commenters with an air of much learnedness, had clearly never open a GR textbook in their lives.
The commenter above seems to believe that local flatness cannot be defined, so perhaps they misinterpreted what you wrote, as you clearly suggest that it is well defined. Also, we are not sure where you got the idea that we present curvature as non-existent for the local observer. This is certainly not the case. Curvature is treated as objective in GR and all observers can measure and agree upon it. But we understand that GR is a complicated topic and that confusions and miscommunications often arise.
@@dialectphilosophy I did not claim that local flatness can't be defined, my hint was that the definition is not obvious - and intuition about it supposedly misleading, and hence should be either clearly defined within a video, or not used.
I did not yet try to find a definition and theorems about it but very much contradicting intuition it seems to me that local flatness only can be broken when the first derivative of the metric (or other function) is not defined or not continuous.
Outstanding video and superbly narrated. Well done!
I learned so much! What stunning visuals !
Great explainatory video!
Loved it!
Simply excellent. Thank you.
This is a very good presentation in the subject. Keep up the good work!
Thank you for posting!!
Elegant, sophisticated, clear and classically modern . Bravo!
Brilliant Work!
Very useful video!
Wow. Your videos are so well done. They are really brillant to get a visual intuition of one of the most fascinating theories of physics.
Thanks a lot!
I am really hoping you will produce more videos.
Absolutely amazing! 🤩🤩
What a great video! Thank you so much 😊🎉❤
Great job. Thank you
My favorite channel now
Also, that introduction has been replayed like a dozen times now, I'm taking notes here man this is amazing content
Thank you so much for your submission! (final video announcement with winners / runner ups out now, by the way)
Keep the great work up!
The best explanation on the internet.
Wow, excellent video!
Thank you for the video.
Best vid of the SpaceTime metric on RUclips
Thumbs up great video!
Good honest information is hard to come by these days. Thank you. It has been a pleasure watching your Uber biased videos.
Amazing videos !! thank you sir !!🤗✌❤
Thanks a lot , Great Illustration.
Superb !
Thank you! After so many books and videos finally somebody explained it perfectly clear and understandable! Impressive work! Better than at any A league university!
Guys, you put things SO brilliantly clear, that even a dumb nuclear physicist like me can understand. thx very much! 👍
Wow nice great work
A dialect video!!!! What a day !!!!
Absolutely incredible
At 1:42, I fell in love with you. This is an amazing video. Very high-quality, clear, and concise, graphical and mathematical projections. My goodness, with videos such as this one, you could teach the derivation of the General Theory of Relativity to 6th graders. I wish it were 3 hours long.
Thank you!
Thank you for the kind words! Glad you enjoyed the video and stayed tuned for the Schwarzschild metric video -- arriving next month!
@@dialectphilosophy Yes, just what I wanted for Christmas!
Thanks for your videos, especially the “why don’t we understand relativity”.
Now you talk about the space-time “distance” between 2 observers to tell that their coordinates systems can be skewed between one another. But indeed I think that you should talk about their relative speed to talk about that, don’t you ?
Hope you’ll continue to do this good job !
Amazing content
Thank you, Dialect!
These videos have been great! Getting an MS in EE I wish I had these descriptions back then. Instead of a pile of matrix equations with almost zero explanation.
Just fantastic!
Nice video.
Phenomenal presentation. Specifically the great combination of beautiful visuals, slow dialogue and supporting mathematics.
Other physics channels don't offer this level of indisputable truth of how our Universe works.
Simply brilliant👏
Thank you for your support!
Excellent explanation and summary. In the end, it's all geometry. More or less easy to visualize, difficult to explain mathematically.
Thank you!
Wonderful videos. Thank you for them. Any suggestion of bibliography which follows the same lines of reasoning but showing all the detais? Regards from Brazil
I honestly thought that I had to get abducted by aliens to get an explanation of the universe this good. Can you please do a video on the Schwarzchild metric? Some of the boring videos on RUclips for that topic are destroying my soul.
Thank you! A video on the Schwarzschild Metric should be arriving sometime in December
@@dialectphilosophy Santa Claus is coming to town! Can not wait! I've been good all year!
This video is too good to be true, thanks!!!
fantastic
Thank you for sharing your understanding, very nice work. Now I wonder, what would be the unit of such spacetime distance ? Does the idea of "unit" even holds here ?
Great question! The unit for spacetime distance is actually just a distance unit (you can notice this when we use light-seconds as the units around the 6:45 mark). If it seems strange that both space and time should have spatial units, it's because in special relativity one uses the spatial displacements of light beams in order to define a measure of time. So in some sense, the spacetime manifold is really a "space-space" manifold.
@@dialectphilosophy Thanks, great answer. I was mainly confused by the dt² - dx² but translating time to distance certainly solves my issue here. Or at least, it displaces the question of unit coherence into a more metaphysical consideration, as per the equations, time and space seem to share the same nature.
Fucking beautiful. Thank you so much. I did a General Relativity course at university and they never taught me this basic fact!
Sir, this is the only time I agree with cursing!!
Bravo👍👍👏👏
Great video glad this showed up in my feed, I'm trying to convince a friend of mine that spacetime exists, and is affected by gravitational potential, and speed of course. Just hard for me to describe when he won't watch videos detailing how it works.
Hopefully it helps! We're going to try to dive deeper into interpretations of what exactly the spacetime manifold really "is" in later videos; we can understand most people's refusal to accept or understand it is based in its highly abstract, removed formulation.
Start with Special Relativity's light clock example of time dilation. Send your friend links to evidence for the Michelson Morley Experiment and subsequent reassuring experiments. Clearly understand how the twin paradox is explained by length contraction reducing relative observers proper time. And describe the Relativity of causality from different perspectives before delving into gravity. The biggest challenge is telling people how and why we live in a block time Universe.
Good luck.
Quoting Stewiesaidthat - Space and Time are two separate frame of reference. Clocks are instruments that measure motion in space. Combining the two frame to believing that clock measures time is what creates the paradox.
Space-Time diagram? That shows one person is experiencing more space in the same amount of time.
Wow I've spent the last month going through Special Relativity in developing the Big Bang Kilonova Hypothesis in order to find simple explanation about how the metric sign flips from negative to positive in the consideration of superluminal velocities. This before showing the importance of how inside a black hole the signs flip and why this so important in well defining the anatomy of a field. Nice one chaps.... don't mind if I do a bit of plagiarism?
Go for it! We're happy to have our work shared. And we're very interested in "superluminal velocities" so make sure to share your work as well!
Very very thankful for this matric tensor linked to GR videos.
Since you have GR your units in light years & light second. But what is the limit in 4D surface matric tensor can be used.
Smaller unit like plank length , is any deviation of geometric symmetry acts?
That's a good question, when exactly does GR as a theory break down? We don't have a great answer to that; certainly by the time you reach the Planck scale it will no longer apply. But generally as you go to smaller scales gravity becomes so weak that GR is negligible anyhow; it would only be in the hypothetical cases of an extremely heavy mass contained in a very tiny area that gravitational effects would be on par with other forces.
I can't wait for you to describe a gravitational field with a metric tensor
We'll be tackling the Christoffel symbols in the next few months, so indeed a GR description of gravitational fields are on their way!
Thanks!
You're welcome! And thank you so much ❤️
I recommend you look at eigenchris series on tensors and general relativity if you are interested in the mathematics involved in GR.
We do love Eigenchris, and have learned more from his channel than any teacher or textbook!
Eigenchris and this channel combined will give birth to a world full of theoretical physicists. Totally awesome! Also, check out the physics explained channel.
Danke!
Gosh, thank you so much for your generosity and your support! ☺ Hope you will enjoy our future content!
Hadn't seen this. Really well explained. Will you maybe cover Christoffel symbols in a future video (like this)?
Christoffel symbols are the next in the series! Unfortunately we've been forced to backtrack a little into special relativity topics, but you should see a video on it in the relatively not-so-distant future.
@@dialectphilosophy Ok. That's ok. I got all the time in the world.
Don't know if you have been reading my comments or not, but uhm. I was very excited when that "time dilation causing gravity nonsense" debunking video came out. I really thought hey, this channel could be something else. And it is really, but I was a bit disappointed and made not such nice comments on videos caming out after that. But I forgot. Your audience uhm .. it's RUclips and I feel like you don't want gaps in your own explanation about relativity. So I regretted being a bit harsh. I believe I changed the comments being milder, because of course you can't rush on and sort of start at I don't know, just picking something, the Kruskal Szekeres coordinates and with that a gravitational singularity.
I actually don't follow any channel anymore, well the science asylum for fun. I think its much more entertainment than education. And I don't know, but have a feeling you really want to educate. That's noble and not easy if you go far beyond popular science of course.
Anyway looking forward to it!👍👍
Hey, great Video as always :)
I am interested in the name of the song that plays during the time period around 9:35. It reminds me of a game i used to play as a child.
All the composition is done in-house; one of our favorite parts in making these videos is trying to come up with music that enhances the viewing (although a lot of viewers find it distracting and get upset by it, so we've considered axing soundtracks altogether). Might throw some of these soundtracks up online if there's any interest.
@@dialectphilosophy Wow that's great. In my opinion the background music does truly enhance the viewing experience as it helps my mind to wander around and to go on the journey it needs to go in order to develop an intuition for the concepts you want to convey. This is what i really appreciate about your channel rather than what i get at university, where it mostly feels as if the professors just focus on calculations and not on intuition and understanding. I don't think the music is too loud either. If you decide to put a collection of soundtracks up online, i would appreciate you letting me know.
Greetings, Kevin
@@Unclouded We're glad to hear that Kevin. We will try to get the music up at some point, and will let you know!
Bruh, thanks 4 the grounding. I'm C, A, T, smart, but this isn't my field. Just a passion 2 entertaining myself since I gave up TV over 3 decades ago. I've been doing brute force calculations w/out. An understanding anchor points in reality. This series is phucking awesome. The meaning of what I have been studying for years has suddenly hit me like the hot kiss of wet fist.
Groovy 2 dah MAX
Dahbig Friendly Satanist, tommyj
How do you make the visuals for this series? Blender?
Finally
How do you explain the difference between atomic clocks from the ground to the sky?
It reminds me of statistics, with the symmetric covariance matrices between coupled random variables. But why is it a symmetric 4x4 matrix? What is the significance and meaning of the off diagonal entries there? I would really expect it to just involve 4 partial derivatives, if gravitational potential is a level function and gravitational field experienced is just the gradient of that, why is there need of a 4x4 matrix in the first place? Though it isn't as crazy as the stress-strain tensor, that's a 4x4 matrix that I would have expected to just be maybe a mass density at that point and a velocity vector but I don't understand at all the role of a 4x4 matrix in describing it.
Of course, for first order effects, you get a vector of partial derivatives, but with 2nd order effects, 2nd derivatives, you get matrices, that's also why you have covariance matrices with random variables, because it's VARIANCES, expected values of the SQUARES of differences between the observed values and means of random variables... so that's just naturally going to happen if it's any 2nd order effect when it is 2 things compared against each other, here spacetime and the MAP which could be any of many things, each with its own metric mapping the real spacetime onto it. But that still doesn't explain the stress strain tensor, all that can possibly be there is still just basically a mass density at that point and where its velocity is taking it, I don't understand how it can make any sense for it to be a 4x4 matrix. There shouldn't be a map of the mass, that the actual mass at that location is being mapped to, right?
To be mathematically precise, the off-diagonal entries of the metric tensor indicate the value of the dot-products between the map coordinate basis vectors as they are expressed in the transformed frame. In most cases, such as in the geometry of spacetime, the dot product of dx and dy will be the same as the dot product of dy and dx, thus half of the diagonal entries become redundant, making the spacetime metric a 4x4 symmetric one.
You are fantastic! This is the most amazing explanation video I have ever seen.
Nice to see somebody posting an accurate review. I don't have anything good to say about this guy, either.
god I've been looking for your explanations for months
I just wish I had an easier way to interact with these principles to internalize them
In 1965 I was 14 and read a book on relativity. It had the Lorentz transform on every page. I understood nothing, but got an A on the book report. I still don't understand it.
@dialectphilosophy But why the minus sign? You said “so it’s hyperbolic” but why must it be hyperbolic? Amazingly clear video overall 👏 but I felt the minus sign explanation was hand wavy
Best till now
@12:50 loved your crisp delivery. In language though, a great teacher, like yourself, can do better. "...stretched or shrunk by the metric tensor's diagonal ..." and "... skewed by an amount determined by the off-diagonal..." --- these phrases can make the younger student think the "metric" is some Aristotlean prime mover or whatever. The metric is only description, not cause. So you say instead, "The lengths measured by calibrated rulers of dx and dy are _described_ by the diagonal components..." Mathematics is description, not ontology. Then you can add, "What _causes_ the surface to be curved are of course electromagnetic, gravitational, and nuclear forces, a whole bunch of them. To compute the metric you need to know about all those forces, or some idealization thereof."
Yes that is much more accurate. You can tell the host is more of a philosophical mathematician than an astrophysicist