Dude, I love your videos. You have no idea how happy you make me. Thank you so much! If you ever come to Germany, I will pay for all your beers and you can stay at my place for free.
Having seen relativity videos from Science Asylum, Veritasium, PBS Spacetime, and even Dr Süskind, I really wanted to SEE the math. And this channel does NOT disappoint! Thanks!!
This is easily the most comprehensive lecture on alternative Schwarzchild coordinates on the internet. Eigenchris explains these difficult concepts in a way that is digestible. A true legend.
8:07 "Far from the Schwarzschild radius, this log part goes to zero ..." But the log|r/rs-1| goes to infinity as r goes infinity. I presume it may be "the log part can be ignored compared with r."
@eigenchris at 8:07 the same problem is happening as in the next video, the log term should goes to positive infinity instead of zero as r approach positive infinity, and the correct description should be the slope (del r) of the log term goes to zero
There's another coordinate system you haven't talked about yet: The GULLSTRAND- PAINLEVÉ coordinates, or GP coordinates for short. Paul PAINLEVÉ (1863-1933) published it in 1921 and Allvar GULLSTRAND (1862-1930), independently from PAINLEVÉ, in 1922. In 1933, George LEMAÎTRE (1894-1966, also known for his work on cosmic expansion) published a paper in which he explicitly showed that GP coordinates are obtained by simple coordinate transform from SCHWARZSCHILD coordinates. I would be glad to see a video on this topic.
35:22 we haven't observed a Schwarzschild white hole, but the big bang singularity should look similiar to the one of Schwarzschild on the Kruskal diagram, though with a slope more spacelike than the 45° so that all time- and lightlike geodesics who can only move with ≤45° would have to originate in it
Unfortunately, no. I'm all done with the relativity videos I plan on making. The channel dXoverdteqprogress has a short video on it: ruclips.net/video/zgG6y5uMjuA/видео.html
@@chacubra I'm planning on doing a video series on spinors. Should start before the end of 2022. And yeah, the clips he plays before his videos are cute/funny.
@@eigenchris Good to hear. Before I committed to GR I spent several months on QFT and advanced QM, so getting more intuition about spinors is neat because my overall goal is quantum gravity.
32:51 hey does that mean inside a black hole's event horizon we can receive light coming from this parallel universe? By the way, thanks for making these videos, they are quite interesting and knowledge dense.
Theoretically, yes (again, this is for eternal black holes). But in practice, anyone who saw such a thing wouldn't be able to report back to us due to the event horizon.
Sorry but I've never studied those. My goal with this series is to cover the basics of GR so that people can learn the rest on their own. I probably won't cover that.
I've heard the Eddington-Finkelstein coordinates involve creating the "analytic continuation" of Schwarzcshild coordinates. This "analytic continuation" is also used in the Riemann Zeta function, as far as I know, is also constructed using "analytic continuation". Unfortunately, I don't really know the details of what that involves, so I can't tell you more.
Awesome vedio!!!! Interconnection are well explained. Please make some vedio on statistics mathematical branch various probability distribution function.
I haven’t finished the video, so this may have already been answered, but Can someone think of the “geometry” of spacetime as being the “metric tensor components”? You say that we can change cooridinates but the geometry of spacetime remains the same, so could this be thought of as the metric tensor components staying the same? (Like how the minkowski metric tensor components stay the same under a Lorentz transformation). EDIT: Nevermind I just saw that the components for the metric for schwartzchild coords vs eddington finkelstein coord are different
It's a bit like taking flat space and chaging from cartesian to polar coordinates. The components of most tensors will change when you change coordinates. The only numbers that will 100% always stay the same in all coordinates are scalars like the Ricci scalar, or the length of vectors. In flat space, the entire Riemann Tensor is zero by definition, so in flat space the Riemann Tensor will always be zero in all coordinates.
29:30 This is wrong. You get 2 answers when you have x² = 9, not when you have x = √9. Square root is a well-defined function with a definitive principal branch. Many instructors fail to distinguish between these two equations and it leads to a lot of confusion later on. Just a minor pet peeve of mine. 35:38 What about the Big Bang? Isn't that a white hole?
Yeah, I've never been very well-versed in analysis. Hoping I got the general point across. Some people have suggested the Big Bang is a white hole but I don't think that's a mainstream position right now.
@@eigenchris You always get the general point across well. Thanks for putting in the work to make these. Not many channels dive so deep into the math, and if you ask me the math is the most important part of science.
@@greenguo1424 No problem. I think my brain and eyes go about 10x more senile trying to triple check all my equations for errors, and sometimes still not catching some after uploading.
So due to gravitational time dilation, it still seems like any observer that exists permanently outside the Schwartshield radius will never observe an object cross the event horizon, even though objects can cross it, and the only way to observe something cross the event horizon is to cross it yourself?
do we need to create more and more coordinate systems until we finally sure we get the right physical insight? (seeing that in schwarzschild coordinate, Rₛ is singular, and incoming/outgoing lightbeams also don't work at Rₛ in EF coordinate)
Another great video! I have a question on the part of white holes and reversed time light rays. Since you said that Einstein-Rosen bridges only considers eternal black holes, and as we have yet no confirmation of their "other end" white holes, neither another universes.. May we consider the lower main diagonal region of KS coordinate system to be only a mirror of the upper main diagonal region, indicating a simetry for antimatter, that is said to be made of time-reversed particles (and light since photons are their own antiparticles)?
Sorry, I missed this comment. The maximally extended coordinate system predicts 2 Flamm's Paraboloids that join up with each other. But as I mention near the end of the video, regions 3 and 4 only exist for eternal black holes. So if a black hole forms as the result of a collapsed star, the parallel region doesn't exist and there is no wormhole.
@@eigenchris Thanks Chris for clear my doubt by your awesome explanation. I am waiting for your next video. What would be your next video topic and when?
@@asrafali8093 The next video is my last video for relativity. It's on cosmological redshift. It should be up within a week. After that I will take a break.
At he end of 108a, you underlined the assumption that the Schwarzschild (exterior) metric is "only valid outside the mass". In this video, is there a specific reason to not use Schwarzschild interior metric going beyond the horizon ? - en.wikipedia.org/wiki/Interior_Schwarzschild_metric
The interior metric refers to the region inside a massive body like a planet or star. For a black holes, there is essentially no "interior" metric, because all the mass is condensed into the singularity. The "interior" of a black hole (inside the event horizon) still uses the "exterior schwarzschild metric", since it's still a vacuum region with zero Ricci Tensor.
@@eigenchris Can we see it as continuous process starting from a star gaining mass ? I would find very interesting if you could consider to create a video about it.
The "interior" metric you linked to assumes the body is a fluid with constant density/pressure. Real starts don't have continuous density, so the metric is a bit idealized. Unfortunately I'm starting to lose the desire to make many more GR videos, as I've been doing them for nearly 2.5 years. I plan on making 3-4 more and then switching to something else.
This is a very basic question and probably shows that I haven't internalized something properly. Is orthogonality a fact about the geometry or of a basis? I had thought that basis vectors are orthogonal by definition, meaning that in a given basis i can see my own basis vectors as orthogonal while another basis may not seem orthogonal to me but will after a coordinate transformation. Or is orthogonality derived from the metric? I ask because in the review section you keep expressing the basis you defined as non-orthogonal in terms of the orthogonal one. But what if I don't have that? Do I have to know the metric? And if it's not diagonal I know my coordinates are not orthogonal? What do I need to review?
"Orthogonality" is a relationship between two vectors, and it requires the vector space to have a "metric", (aka "dot product") in order to define it. The definition of two vectors "v" and "u" being orthogonal is that their dot product is zero. Without the dot product (aka metric) defined, you can't define what orthogonality means. So you can take the same point in the same geometry, but put different basis vectors on it. Some basis vectors might be orthogonal, others might not be. The definitions of a "basis" that I've seen usually require "Linear Independence" between the basis vectors, but not "Orthogonality" among the basis vectors. I tried to drive this point home by showing you can use a non-orthogonal basis to measure vectors without any problems. The metric tensor components will tell you if your basis is orthogonal or not. If it's diagonal, all the basis vectors are mutually orthogonal. Any non-zero off-diagonal component indicates a pair of basis vectors that are non-orthogonal. To make any progress in differential geometry, you have to be given a metric to start with somehow. You can then change coordinates and see how the components of that metric change. But the metric has to be given to you to start with. Maybe go back and watch my "Tensors for Beginners 9" video? I tried to explain the metric as simply as possible in that one. I also repeat many of the same points in "Relativity 103d", but with more of a focus on matrices instead of Einstein index notation.
@@eigenchris Thank you very much, I'll check that video out again. Maybe I am just conflating linear independence and orthogonality! So linear independence means they are not linear combinations of each other. That is _not_ a fact of the geometry, because in a different basis they might be. Orthogonal means their dot product is zero. This is a fact of the geometry and basis independent. Is this correct? I had taken orthogonal to mean the first, so that's my mistake.
@mica ge by choosing a new basis in which one of them is a basis vector and the other isn't. Now the second one is a linear combination that includes the first. So it is not linearly independent of the first. Or did I misunderstand linear independence as well?
I have a request to you. I've been learning more with your videos than any other source of information. Could u make the same video u did here but with another space-time. For exemple, u could derive de EF and KS coordinates for Reissner-Nordström metric.
Sorry, I've never studied that metric, and I'm done with making relativity videos. I know the Dietterich Labs channel has a video on this. Maybe you can give that one a try.
Another question. You mention that white holes and the parallel region are a possibility if there is an eternal black hole. Why must there be an eternal black hole for this possibility to manifest? Also, if there is an eternal black hole, does that *guarantee* that there are white holes and a “parallel” universe? Or is it still a possibility?
you dont really have guarantees in physics (better to say reality) especially regarding parallel universes. to your first question I'm not sure but I'm dubious about it, possibly related to energy conservation
As the above comment says, I don't think we have any guarantees. With theoretical physics, it can be hard to tell if an equation is telling us something true, or just has some weird math behaviour when it stops working. In the past there have been a number of cases where an equation "going wrong" turned out to lead to true physics (black holes and anti-matter are two big examples), but at the end of the day, only experiment can verify things.
It must be an eternal black hole because if it's not and it had formed due to the collapse of a star then regions 3 and 4 are occupied by that star PS: I learned this from Sean Caroll's channel
10:00 ff: The infalling light beams and observer's world lines being connected at infinity in a remote observer's coordinate time, what about the BH evaporation within _finite_ coordinate time? Shouldn't the infalling observer see the BH evaporate much faster?
Hi Chris. I have to say that your videos are a very good source on GR and for that thank you. I would like to ask that when we are choosing KS coordinates we first choose u and v as possible coordinates. Is there a rigorous reason why we choose them other than them individually eliminating the artificial singularity at r=rs and they may provide some promising results? Thank you for your time in advance.
I believe that was the main reason for the invention of the KS coordinates. There is a related coordinate system called the "Penrose Diagram" where beams of light are also diagonal lines, and it is sometimes used to visualize black holes, white holes, and wormholes. But I haven't studied those. My memory is a big foggy but I think either Kruskal or Szekeres just really loved playing with coordinate systems and invented the KS coordinates through trial and error.
Here's a question: if a distant observer never sees something cross the event horizon in finite time, what happens when the black hole evaporates in finite time? Does it mean nothing ever does cross? Or do we have to start diving into subtleties of event horizons versus trapped surfaces and suchlike?
That's an interesting question. I've never learned about Hawking radiation so I can't give a proper answer. This video largely considers the case of an "eternal" black hole and ignores Hawking radiation. As you say, handling this properly might involve objects being "trapped" very close to the event horizon, and could possibly escape after the black hole evaporates. I can't really confirm if this is true though. You might be interested in this answer on Physics Stack Exchange: physics.stackexchange.com/questions/21319/how-can-anything-ever-fall-into-a-black-hole-as-seen-from-an-outside-observer
@@eigenchris Thank you! This discussion is quite relevant! I'm still reading through it. I'm glad to find out that, although it might not have a "settled" answer, my question seems to be a good one. Thanks again!
I have heard of the proposition that the known universe is the inside of a black hole and our experience of time only moving forward is equivalent to falling to the center. With your understanding, does that seem plausible? If so, why would we perceive the universe as expanding?
@@eigenchris according to the "soft" science I've read and is mentioned in other videos I've seen, it seems that, someone compared the estimate mass of the visible universe to it's size , and found the schwazrtschild radius to be bigger than the visible universe‽ That's the main "evidence" the other is comparing the "red limit" to an event horizon, but the latter seems too superficial to me. I've asked if the size used to calculate was based on the 14.7± billion light-years as seen , or the ~40-96± billion wide it should be by now accounting for when the light left and purported expansion rate, but never got an answer. If the current visible universe is not within it's own schwazrstchild radius now it seems likely it was in the past. If it is now, I want to know how long till it's not, and what might happen at that time. The concept as a whole appeals to me because (assuming passage through the event horizon is equivalent to a zero crossing of the time dimension) I can imagine that that point of zero time that everything would smoothly pass through would look like, and effectively be cuz you can't see beyond, the beginning of time from within, also whatever falls into an event horizon would be torn down to its most elementary level and be in it's maximum compressed state and hence resemble the models of the beginning of the universe. Another thing I've heard said a lot in pop-sci is that Time and space reverse roles on the inside of a black hole with it being explained as the passage of time being equivalent to the path towards the singularity. Here is a question, (assuming my presumption that the inside of an event horizon would be perceived as a single zero point crossing beginning of time from within the horizon is correct) wouldn't the "approach to Infinity" that moving towards the singularity that the "infinity" represents also appear like a universe expanding toward infinite size? Also, Following the above "logic" would the decay of all matter followed by heat death of the universe be equivalent to last bit of energy evaporating from a dieing black hole? Essentially when the event horizon finally cought up to the singularity? ....As if the beginning and end of time is at the event horizon?
Hello Chris, around 08:00 you say the solution of dct/dr leads to ct(r)=+-(r+r_s log (r/r_s - 1)+k. when i solve it gives me ct(r)=+-r+_s log(r-r_s)+k which can also be found in other textbooks. Can you explain the missing factor 1/r_s in the argument of log? Thanks!
I believe you can do this algebraic manipulation: log(r/rs - 1) = log((r-rs)/rs) = log(r-rs) - log(rs) That last "-log(rs)" term is a constant that can be absorbed into the "+k". So the two are equivalent.
I'm not aware of any specific motivation. I think the person who discovered these coordinates was just trying various things until they found something that worked.
What would happen to two observers free-falling towards the event horizon together, where one observer is staggered slightly behind the other in the radial direction. Would they still observe the event horizon at the same place? Or would they pass through the event horizon together? And if so, how far apart would they have to be to not be able to pass through the horizon together?
That's an interesting question... off the top of my head, without doing a calculation, it feels like they might have to pass through the event horizon together, since anything outside the horizon can't see anything pass through the horizon. If I find the answer I'll add another comment.
They will cross at the same time at exactly t=infinity for all frames outside the event horizon, but at their own expected proper time. If there is a mirror 1 light year away from the black hole and you send a photon out to hit it before you cross the event horizon, it will hit the mirror and come back to you before you cross no matter how close you were to the event horizon when you sent it.
Dear Chris I have seen most of your vedio it take me to a different level of thinking and imagination. Thank you so much!!! Since we have not arrived to concept of negative gravitional force as of now we cannot conceptualize white hole. As per electromagnetic theory we have concept of positive and negative charges also we have dipoles. But still only the concept of monopole exist. Physically difficult to seperate in our space time. I strongly belive there should negative gravtional force to nullify the equilbrium of space time. But as per basic elementary particle of physic we have basic matter and force particles. But do we have fundamental particle connecting them. Another dimension For sake of our understanding let think 2-2=0 ; 5-5=0 also 1000-1000=0 is also zero. But in physical world this nullification take matter force and time or scalar vector or tensor to arrive at zero. Mathematically left and side equal to right hand side but mere right hand side result cannot conculde the zero. According to me concept of zero or sunya. are a bit different.
The most straightforward visual explanation is just looking at the spacetime diagram and seeing that horizontal lines are eventually interrupted by the hyperbolia event horizons on either side... the point where the horizontal line is tangent to the singularity is when the pinching off happens. If you want the mathematical proof, you would have to compute the proper lengths for horizontal lines other than T=0, using the KS metric. The Flamm's Paraboloid formula I should works for the case of ct = 0, which is also the same horizontal line as T=0, so I can use the shortcut of using the Schwarzschild metric instead, which is easier. For the T ≠ 0 case the KS formula ends up being more complicated, but it can be done.
Ok amazing, i should have understood. Thank you for the answer and for both the relativity and the differential geometry playlists! To be fair I understood more watching your videos than following my phd's professor, my compliments to you!
@@worship_the_sun_officialch506 Thanks. I considered talking about that more, but I try to keep my videos under 30 minutes, and this one was already at 37, so I cut it out.
wonder... do you have a personal idea on what kind of matter can create a white hole? black holes themselves used to be treated as bizarre mathematical curiosity, and now we have a telescope image
If you had a contact email address I would send you this information by email. Since I can't contact you that way, I'll explain my theory based on GR here: 4-space, spacetime coordinates are technically specified as (ct, x, y, z) or (ct, x1, x2, x3) not (t, x, y, z) or (t, x1, x2, x3). Since ct is the speed of light multiplied by time and has units of distance - not time - this means the x0 coordinate of the spacetime point (x0,x1,x2,x3) is a distance - - i.e. meters - not a time in seconds. Implications of this fact: Let A and B define spacetime interval dS, using Schwarzchild metric, that describes Earth's gravity to a high degree of accuracy: A(x0,x1,x2,x3) = initial event B(x0,x1,x2,x3) = final event Assume A and B are inside a fresh water lake. x0 refers to photons initially radiated from A at location of the initial event in the water. They travel to point B at location of final event in the water. Assume A is coordinate system origin. Coordinate ct of B(ct,x1,x2,x3) is the distance photons traveled within interval dS. The light photons initially radiated from point A ARE TRAVELING THROUGH WATER So x0, the distance traveled by the photon = ct is THE SPEED OF LIGHT THROUGH WATER multiplied by t. The speed of light through water is slower than speed of light in vacuum: the speed of light through water is 0.75c. So for coordinate x0 of B (x0,x1,x2,x3), (ct,x1,x2,x3) the value of c should be speed of light in water - not the speed of light in a vacuum. Since the GR field equation is based on the same (ct,x1,x2,x3) coordinates, the value of c in the GR field equation (c to 4th power) must, therefore, be modified to equal THE SPEED OF LIGHT IN THE MEDIUM where these coordinates are located. IMPLICATIONS: Lene Hau at Harvard University showed that a Bose-EinsteinCondensate (BEC) reduces the speed of light by many orders of magnitude; and if the BEC is pure enough the speed of light is reduced to 0. The GR field equation shows (as specified in stress-energy-momentum tensor) i.imgur.com/eGX7PST.png ... that the mass-the energy density, energy flux, momentum density, shear stress, or pressure needed to create a gravitational field; or the negative pressureTension needed to create an anti-gravitational field, is proportional to c to 4th power: i.imgur.com/vkjvgyQ.jpg shifting terms: i.imgur.com/YaNeiFc.jpg Therefore within a BEC where the speed of light is reduced by many orders of magnitude the modified value of c in the GR field equation shows that the mass-energy density, energy flux, momentum density, shear stress, or pressure needed to create a gravitational field, or negative pressureTension needed to create an anti-gravitational field is also reduced by many orders of magnitude. This means that it is theoretically possible to create artificial gravitational or anti-gravitational field in a laboratory setting. If you are interested, let me know, and I will explain how an anti-gravitational field can be achieved with an electric field - that creates electric dipole Tension in the atoms of a material.
Be careful with the words "no reason to believe." Talking about observational evidence is one thing, but surely you believe in more things than strictly the things you're able to observe!
Dude, I love your videos. You have no idea how happy you make me. Thank you so much! If you ever come to Germany, I will pay for all your beers and you can stay at my place for free.
I can't imagine how much work goes into these videos. This series is incredible. THANK YOU!
Having seen relativity videos from Science Asylum, Veritasium, PBS Spacetime, and even Dr Süskind, I really wanted to SEE the math. And this channel does NOT disappoint! Thanks!!
This is easily the most comprehensive lecture on alternative Schwarzchild coordinates on the internet. Eigenchris explains these difficult concepts in a way that is digestible. A true legend.
As an aspiring physicist, this series has been one of the best things that’s happened to me! Thanks so much!
Glad you're finding it helpful!
The r* part was not necessary. The partial derivative of ct with respect to r_in is the same as dct/dr, which is calculated at 7:53.
8:07 "Far from the Schwarzschild radius, this log part goes to zero ..." But the log|r/rs-1| goes to infinity as r goes infinity. I presume it may be "the log part can be ignored compared with r."
Wow. This is the ultimate channel to understand general relativity.
@eigenchris at 8:07 the same problem is happening as in the next video, the log term should goes to positive infinity instead of zero as r approach positive infinity, and the correct description should be the slope (del r) of the log term goes to zero
There's another coordinate system you haven't talked about yet: The GULLSTRAND- PAINLEVÉ coordinates, or GP coordinates for short. Paul PAINLEVÉ (1863-1933) published it in 1921 and Allvar GULLSTRAND (1862-1930), independently from PAINLEVÉ, in 1922.
In 1933, George LEMAÎTRE (1894-1966, also known for his work on cosmic expansion) published a paper in which he explicitly showed that GP coordinates are obtained by simple coordinate transform from SCHWARZSCHILD coordinates.
I would be glad to see a video on this topic.
35:22 we haven't observed a Schwarzschild white hole, but the big bang singularity should look similiar to the one of Schwarzschild on the Kruskal diagram, though with a slope more spacelike than the 45° so that all time- and lightlike geodesics who can only move with ≤45° would have to originate in it
Exceptional my man. Im going to write my thesis about the reissner-nordstrom metric, are you intending to make a video about it?
Unfortunately, no. I'm all done with the relativity videos I plan on making. The channel dXoverdteqprogress has a short video on it: ruclips.net/video/zgG6y5uMjuA/видео.html
@@eigenchris what's next? And thanks a lot for introducing me to that channel, such a unique/ humorous way of presenting the material
@@chacubra I'm planning on doing a video series on spinors. Should start before the end of 2022. And yeah, the clips he plays before his videos are cute/funny.
@@eigenchris Good to hear. Before I committed to GR I spent several months on QFT and advanced QM, so getting more intuition about spinors is neat because my overall goal is quantum gravity.
32:51 hey does that mean inside a black hole's event horizon we can receive light coming from this parallel universe?
By the way, thanks for making these videos, they are quite interesting and knowledge dense.
Theoretically, yes (again, this is for eternal black holes). But in practice, anyone who saw such a thing wouldn't be able to report back to us due to the event horizon.
Great video !!! After this series can you do a tutorial on singularity theorems and the Raychaudhuri equation ?
Sorry but I've never studied those. My goal with this series is to cover the basics of GR so that people can learn the rest on their own. I probably won't cover that.
this series has been running since 2 years, why would you even wait for it to end lol
The Eddigton-Finkelstein 'Map' reminds me at the Riemann-Zeta-Function and it's 'magic numberline' at 0,5.
I've heard the Eddington-Finkelstein coordinates involve creating the "analytic continuation" of Schwarzcshild coordinates. This "analytic continuation" is also used in the Riemann Zeta function, as far as I know, is also constructed using "analytic continuation". Unfortunately, I don't really know the details of what that involves, so I can't tell you more.
Awesome vedio!!!! Interconnection are well explained. Please make some vedio on statistics mathematical branch various probability distribution function.
I haven’t finished the video, so this may have already been answered, but Can someone think of the “geometry” of spacetime as being the “metric tensor components”? You say that we can change cooridinates but the geometry of spacetime remains the same, so could this be thought of as the metric tensor components staying the same? (Like how the minkowski metric tensor components stay the same under a Lorentz transformation).
EDIT: Nevermind I just saw that the components for the metric for schwartzchild coords vs eddington finkelstein coord are different
It's a bit like taking flat space and chaging from cartesian to polar coordinates. The components of most tensors will change when you change coordinates. The only numbers that will 100% always stay the same in all coordinates are scalars like the Ricci scalar, or the length of vectors. In flat space, the entire Riemann Tensor is zero by definition, so in flat space the Riemann Tensor will always be zero in all coordinates.
29:30 This is wrong. You get 2 answers when you have x² = 9, not when you have x = √9. Square root is a well-defined function with a definitive principal branch. Many instructors fail to distinguish between these two equations and it leads to a lot of confusion later on. Just a minor pet peeve of mine.
35:38 What about the Big Bang? Isn't that a white hole?
Yeah, I've never been very well-versed in analysis. Hoping I got the general point across.
Some people have suggested the Big Bang is a white hole but I don't think that's a mainstream position right now.
@@eigenchris You always get the general point across well. Thanks for putting in the work to make these. Not many channels dive so deep into the math, and if you ask me the math is the most important part of science.
10:13, since ln of negative numbers aren't real, how can we "pull down" the ct coordinate values when inside event horizon? :)
There's an absolute value inside the ln operation, so there's no negative inputs.
@@eigenchris It's official. Both my brain and eyes have gone senile. Thanks for clarifying!
@@greenguo1424 No problem. I think my brain and eyes go about 10x more senile trying to triple check all my equations for errors, and sometimes still not catching some after uploading.
@@eigenchris Haha please don't go. To err is human. This is already the best modern physics class anywhere. We need you! Let me buy you a kofi.
So due to gravitational time dilation, it still seems like any observer that exists permanently outside the Schwartshield radius will never observe an object cross the event horizon, even though objects can cross it, and the only way to observe something cross the event horizon is to cross it yourself?
do we need to create more and more coordinate systems until we finally sure we get the right physical insight? (seeing that in schwarzschild coordinate, Rₛ is singular, and incoming/outgoing lightbeams also don't work at Rₛ in EF coordinate)
try it and let us know
Someone in a parallel Universe: There's no evidence that black holes exist yet
For integral at 8:00 should be have other term( -Rs).
Sorry, I don't follow. What do you mean?
Ct=r+RsLog(r/RS -1) -Rs
I'd think the last term is free to be any number, since it's just an integration constant.
I've got your point. Thank you
Another great video!
I have a question on the part of white holes and reversed time light rays. Since you said that Einstein-Rosen bridges only considers eternal black holes, and as we have yet no confirmation of their "other end" white holes, neither another universes.. May we consider the lower main diagonal region of KS coordinate system to be only a mirror of the upper main diagonal region, indicating a simetry for antimatter, that is said to be made of time-reversed particles (and light since photons are their own antiparticles)?
Watching all from EIGENCHRIS, and finally getting it. Books are so hard to wade through.
Chris, does Flamm's Paraboloid predict that worm hole connect two diffrent places in space-time?
Sorry, I missed this comment. The maximally extended coordinate system predicts 2 Flamm's Paraboloids that join up with each other. But as I mention near the end of the video, regions 3 and 4 only exist for eternal black holes. So if a black hole forms as the result of a collapsed star, the parallel region doesn't exist and there is no wormhole.
@@eigenchris Thanks Chris for clear my doubt by your awesome explanation.
I am waiting for your next video. What would be your next video topic and when?
@@asrafali8093 The next video is my last video for relativity. It's on cosmological redshift. It should be up within a week. After that I will take a break.
20:15 how d/dv (and d/du) is considered as ingoing(outgoing) light beam as the diagram dont suggerst?
At he end of 108a, you underlined the assumption that the Schwarzschild (exterior) metric is "only valid outside the mass". In this video, is there a specific reason to not use Schwarzschild interior metric going beyond the horizon ? - en.wikipedia.org/wiki/Interior_Schwarzschild_metric
The interior metric refers to the region inside a massive body like a planet or star. For a black holes, there is essentially no "interior" metric, because all the mass is condensed into the singularity. The "interior" of a black hole (inside the event horizon) still uses the "exterior schwarzschild metric", since it's still a vacuum region with zero Ricci Tensor.
@@eigenchris Can we see it as continuous process starting from a star gaining mass ? I would find very interesting if you could consider to create a video about it.
The "interior" metric you linked to assumes the body is a fluid with constant density/pressure. Real starts don't have continuous density, so the metric is a bit idealized. Unfortunately I'm starting to lose the desire to make many more GR videos, as I've been doing them for nearly 2.5 years. I plan on making 3-4 more and then switching to something else.
This is a very basic question and probably shows that I haven't internalized something properly.
Is orthogonality a fact about the geometry or of a basis? I had thought that basis vectors are orthogonal by definition, meaning that in a given basis i can see my own basis vectors as orthogonal while another basis may not seem orthogonal to me but will after a coordinate transformation.
Or is orthogonality derived from the metric?
I ask because in the review section you keep expressing the basis you defined as non-orthogonal in terms of the orthogonal one. But what if I don't have that? Do I have to know the metric? And if it's not diagonal I know my coordinates are not orthogonal?
What do I need to review?
"Orthogonality" is a relationship between two vectors, and it requires the vector space to have a "metric", (aka "dot product") in order to define it. The definition of two vectors "v" and "u" being orthogonal is that their dot product is zero. Without the dot product (aka metric) defined, you can't define what orthogonality means. So you can take the same point in the same geometry, but put different basis vectors on it. Some basis vectors might be orthogonal, others might not be. The definitions of a "basis" that I've seen usually require "Linear Independence" between the basis vectors, but not "Orthogonality" among the basis vectors. I tried to drive this point home by showing you can use a non-orthogonal basis to measure vectors without any problems.
The metric tensor components will tell you if your basis is orthogonal or not. If it's diagonal, all the basis vectors are mutually orthogonal. Any non-zero off-diagonal component indicates a pair of basis vectors that are non-orthogonal. To make any progress in differential geometry, you have to be given a metric to start with somehow. You can then change coordinates and see how the components of that metric change. But the metric has to be given to you to start with.
Maybe go back and watch my "Tensors for Beginners 9" video? I tried to explain the metric as simply as possible in that one. I also repeat many of the same points in "Relativity 103d", but with more of a focus on matrices instead of Einstein index notation.
@@eigenchris Thank you very much, I'll check that video out again. Maybe I am just conflating linear independence and orthogonality!
So linear independence means they are not linear combinations of each other. That is _not_ a fact of the geometry, because in a different basis they might be.
Orthogonal means their dot product is zero. This is a fact of the geometry and basis independent.
Is this correct?
I had taken orthogonal to mean the first, so that's my mistake.
@mica ge by choosing a new basis in which one of them is a basis vector and the other isn't. Now the second one is a linear combination that includes the first. So it is not linearly independent of the first.
Or did I misunderstand linear independence as well?
I have a request to you. I've been learning more with your videos than any other source of information. Could u make the same video u did here but with another space-time. For exemple, u could derive de EF and KS coordinates for Reissner-Nordström metric.
Sorry, I've never studied that metric, and I'm done with making relativity videos. I know the Dietterich Labs channel has a video on this. Maybe you can give that one a try.
Another question. You mention that white holes and the parallel region are a possibility if there is an eternal black hole. Why must there be an eternal black hole for this possibility to manifest? Also, if there is an eternal black hole, does that *guarantee* that there are white holes and a “parallel” universe? Or is it still a possibility?
you dont really have guarantees in physics (better to say reality) especially regarding parallel universes. to your first question I'm not sure but I'm dubious about it, possibly related to energy conservation
As the above comment says, I don't think we have any guarantees. With theoretical physics, it can be hard to tell if an equation is telling us something true, or just has some weird math behaviour when it stops working. In the past there have been a number of cases where an equation "going wrong" turned out to lead to true physics (black holes and anti-matter are two big examples), but at the end of the day, only experiment can verify things.
It must be an eternal black hole because if it's not and it had formed due to the collapse of a star then regions 3 and 4 are occupied by that star
PS: I learned this from Sean Caroll's channel
10:00 ff: The infalling light beams and observer's world lines being connected at infinity in a remote observer's coordinate time, what about the BH evaporation within _finite_ coordinate time?
Shouldn't the infalling observer see the BH evaporate much faster?
Hi Chris. I have to say that your videos are a very good source on GR and for that thank you. I would like to ask that when we are choosing KS coordinates we first choose u and v as possible coordinates. Is there a rigorous reason why we choose them other than them individually eliminating the artificial singularity at r=rs and they may provide some promising results? Thank you for your time in advance.
I believe that was the main reason for the invention of the KS coordinates. There is a related coordinate system called the "Penrose Diagram" where beams of light are also diagonal lines, and it is sometimes used to visualize black holes, white holes, and wormholes. But I haven't studied those.
My memory is a big foggy but I think either Kruskal or Szekeres just really loved playing with coordinate systems and invented the KS coordinates through trial and error.
@@eigenchris OK, so it is an informed guess. Thank you again.
Just a small note at about 30:00 that the square root of a positive real number is only the positive root haha, so that argument does not really work!
That's more of a notational definition. The mathematical idea of therr being 2 answers is still true.
and if a massive object approaches a white hole from infinity, would it feel some sort of repulsion?
Here's a question: if a distant observer never sees something cross the event horizon in finite time, what happens when the black hole evaporates in finite time? Does it mean nothing ever does cross? Or do we have to start diving into subtleties of event horizons versus trapped surfaces and suchlike?
That's an interesting question. I've never learned about Hawking radiation so I can't give a proper answer. This video largely considers the case of an "eternal" black hole and ignores Hawking radiation. As you say, handling this properly might involve objects being "trapped" very close to the event horizon, and could possibly escape after the black hole evaporates. I can't really confirm if this is true though. You might be interested in this answer on Physics Stack Exchange: physics.stackexchange.com/questions/21319/how-can-anything-ever-fall-into-a-black-hole-as-seen-from-an-outside-observer
@@eigenchris Thank you! This discussion is quite relevant! I'm still reading through it. I'm glad to find out that, although it might not have a "settled" answer, my question seems to be a good one. Thanks again!
I need the easiest way to derive the Kerr metric.
I have heard of the proposition that the known universe is the inside of a black hole and our experience of time only moving forward is equivalent to falling to the center.
With your understanding, does that seem plausible? If so, why would we perceive the universe as expanding?
I've never heard of this. What would be the evidence for it?
@@eigenchris according to the "soft" science I've read and is mentioned in other videos I've seen, it seems that, someone compared the estimate mass of the visible universe to it's size , and found the schwazrtschild radius to be bigger than the visible universe‽
That's the main "evidence" the other is comparing the "red limit" to an event horizon, but the latter seems too superficial to me.
I've asked if the size used to calculate was based on the 14.7± billion light-years as seen , or the ~40-96± billion wide it should be by now accounting for when the light left and purported expansion rate, but never got an answer.
If the current visible universe is not within it's own schwazrstchild radius now it seems likely it was in the past. If it is now, I want to know how long till it's not, and what might happen at that time.
The concept as a whole appeals to me because (assuming passage through the event horizon is equivalent to a zero crossing of the time dimension) I can imagine that that point of zero time that everything would smoothly pass through would look like, and effectively be cuz you can't see beyond, the beginning of time from within, also whatever falls into an event horizon would be torn down to its most elementary level and be in it's maximum compressed state and hence resemble the models of the beginning of the universe.
Another thing I've heard said a lot in pop-sci is that Time and space reverse roles on the inside of a black hole with it being explained as the passage of time being equivalent to the path towards the singularity.
Here is a question, (assuming my presumption that the inside of an event horizon would be perceived as a single zero point crossing beginning of time from within the horizon is correct) wouldn't the "approach to Infinity" that moving towards the singularity that the "infinity" represents also appear like a universe expanding toward infinite size?
Also,
Following the above "logic" would the decay of all matter followed by heat death of the universe be equivalent to last bit of energy evaporating from a dieing black hole? Essentially when the event horizon finally cought up to the singularity?
....As if the beginning and end of time is at the event horizon?
@@eigenchris never mind my musing, what do you think about the mass vs radius argument?
@@eigenchris ruclips.net/video/jeRgFqbBM5E/видео.html
Hello Chris, around 08:00 you say the solution of dct/dr leads to ct(r)=+-(r+r_s log (r/r_s - 1)+k. when i solve it gives me ct(r)=+-r+_s log(r-r_s)+k which can also be found in other textbooks. Can you explain the missing factor 1/r_s in the argument of log? Thanks!
I believe you can do this algebraic manipulation: log(r/rs - 1) = log((r-rs)/rs) = log(r-rs) - log(rs)
That last "-log(rs)" term is a constant that can be absorbed into the "+k". So the two are equivalent.
@@eigenchris that sounds great, thanks for your efforts!
Is there a motivation for the rescaling in KS coordinates beyond "this just happens to work"?
I'm not aware of any specific motivation. I think the person who discovered these coordinates was just trying various things until they found something that worked.
What would happen to two observers free-falling towards the event horizon together, where one observer is staggered slightly behind the other in the radial direction. Would they still observe the event horizon at the same place? Or would they pass through the event horizon together? And if so, how far apart would they have to be to not be able to pass through the horizon together?
That's an interesting question... off the top of my head, without doing a calculation, it feels like they might have to pass through the event horizon together, since anything outside the horizon can't see anything pass through the horizon. If I find the answer I'll add another comment.
@@eigenchris appreciate it thank you!
They will cross at the same time at exactly t=infinity for all frames outside the event horizon, but at their own expected proper time. If there is a mirror 1 light year away from the black hole and you send a photon out to hit it before you cross the event horizon, it will hit the mirror and come back to you before you cross no matter how close you were to the event horizon when you sent it.
Dear Chris
I have seen most of your vedio it take me to a different level of thinking and imagination. Thank you so much!!!
Since we have not arrived to concept of negative gravitional force as of now we cannot conceptualize white hole.
As per electromagnetic theory we have concept of positive and negative charges also we have dipoles. But still only the concept of monopole exist.
Physically difficult to seperate in our space time.
I strongly belive there should negative gravtional force to nullify the equilbrium of space time.
But as per basic elementary particle of physic we have basic matter and force particles. But do we have fundamental particle connecting them.
Another dimension
For sake of our understanding let think 2-2=0 ; 5-5=0 also 1000-1000=0 is also zero. But in physical world this nullification take matter force and time or scalar vector or tensor to arrive at zero. Mathematically left and side equal to right hand side but mere right hand side result cannot conculde the zero.
According to me concept of zero or sunya. are a bit different.
making up coordinate systems to get new insights sounds like an artform
Yeah, I think it is. I don't think KS coordinates were invented until decades after the Schwarzschild solution was found.
May i ask how can you prove/see that the wormhole/Einstein-penrose bridge shrinks as we get further away from the constant time surface T = 0?
The most straightforward visual explanation is just looking at the spacetime diagram and seeing that horizontal lines are eventually interrupted by the hyperbolia event horizons on either side... the point where the horizontal line is tangent to the singularity is when the pinching off happens.
If you want the mathematical proof, you would have to compute the proper lengths for horizontal lines other than T=0, using the KS metric. The Flamm's Paraboloid formula I should works for the case of ct = 0, which is also the same horizontal line as T=0, so I can use the shortcut of using the Schwarzschild metric instead, which is easier. For the T ≠ 0 case the KS formula ends up being more complicated, but it can be done.
Ok amazing, i should have understood. Thank you for the answer and for both the relativity and the differential geometry playlists!
To be fair I understood more watching your videos than following my phd's professor, my compliments to you!
@@worship_the_sun_officialch506 Thanks. I considered talking about that more, but I try to keep my videos under 30 minutes, and this one was already at 37, so I cut it out.
Awesome videos! :D
Your the greatest 👏👏👏
Superb 🤩🤩
wonder... do you have a personal idea on what kind of matter can create a white hole? black holes themselves used to be treated as bizarre mathematical curiosity, and now we have a telescope image
yay new vid!!!!!
„Co-ordinate systems dont change real physics“.
Damn bro we are all schwarzchild's schwarzchildren
Thank you🌹 🌉
If you had a contact email address I would send you this information by email. Since I can't contact you that way, I'll explain my theory based on GR here:
4-space, spacetime coordinates are technically specified as
(ct, x, y, z) or (ct, x1, x2, x3)
not
(t, x, y, z) or (t, x1, x2, x3).
Since ct is the speed of light multiplied by time and has units of distance - not time - this means the x0 coordinate of the spacetime point (x0,x1,x2,x3) is a distance - - i.e. meters - not a time in seconds.
Implications of this fact:
Let A and B define spacetime interval dS, using Schwarzchild metric, that describes Earth's gravity to a high degree of accuracy:
A(x0,x1,x2,x3) = initial event
B(x0,x1,x2,x3) = final event
Assume A and B are inside a fresh water lake.
x0 refers to photons initially radiated from A at location of the initial event in the water. They travel to point B at location of final event in the water. Assume A is coordinate system origin. Coordinate ct of B(ct,x1,x2,x3) is the distance photons traveled within interval dS.
The light photons initially radiated from point A
ARE TRAVELING THROUGH WATER
So x0, the distance traveled by the photon = ct
is THE SPEED OF LIGHT THROUGH WATER multiplied by t.
The speed of light through water is slower than speed of light in vacuum: the speed of light through water is 0.75c. So for coordinate x0 of B (x0,x1,x2,x3), (ct,x1,x2,x3) the value of c should be speed of light in water - not the speed of light in a vacuum.
Since the GR field equation is based on the same (ct,x1,x2,x3) coordinates,
the value of c in the GR field equation (c to 4th power) must, therefore, be modified to equal
THE SPEED OF LIGHT IN THE MEDIUM
where these coordinates are located.
IMPLICATIONS:
Lene Hau at Harvard University showed that a Bose-EinsteinCondensate (BEC) reduces the speed of light by many orders of magnitude; and if the BEC is pure enough the speed of light is reduced to 0. The GR field equation shows (as specified in stress-energy-momentum tensor)
i.imgur.com/eGX7PST.png
... that the mass-the energy density, energy flux, momentum density, shear stress, or pressure needed to create a gravitational field; or the negative pressureTension needed to create an anti-gravitational field,
is proportional to c to 4th power:
i.imgur.com/vkjvgyQ.jpg
shifting terms:
i.imgur.com/YaNeiFc.jpg
Therefore within a BEC
where the speed of light is reduced by many orders of magnitude
the modified value of c in the GR field equation shows that the mass-energy density, energy flux, momentum density, shear stress, or pressure needed to create a gravitational field, or negative pressureTension needed to create an anti-gravitational field
is also reduced by many orders of magnitude.
This means that it is theoretically possible to create artificial gravitational or anti-gravitational field in a laboratory setting.
If you are interested, let me know, and I will explain how an anti-gravitational field can be achieved with an electric field - that creates electric dipole Tension in the atoms of a material.
Be careful with the words "no reason to believe." Talking about observational evidence is one thing, but surely you believe in more things than strictly the things you're able to observe!
Veritasium brought me here
The content is excellent but by god you talk like you learnt to speak from the google text-to-speech