00:00We live in a curved space-time, Einstein tells us, but what curves it and especially how?
00:06Watch the video until the end to discover Einstein's answer and his second famous equation.
00:11Master panda!
00:15In the physics lesson series we've already started in the playlist,
00:19we saw that the space-time we live in is curved, having a precise geometric structure described by its metric
00:25G.
00:26It tells us how time will pass on our clocks, what the distance between objects will be,
00:31as well as the motion of a freely falling body or a ray of light.
00:35However, an essential point is the following.
00:38Who curves space-time and how is it done?
00:41Because until now we have been the ones who build the metric in the few examples mentioned.
00:46Who is responsible for the curvature of the universe we live in?
00:51No, don't think about a creator.
00:53We remain within the framework of physics and here we will look for the answer.
00:58To do this, imagine a universe with two bodies.
01:01One massive and the other small, a test body.
01:04In Newton's explanation, the massive body attracts the small one and thus sets it in motion.
01:10In Einstein's explanation, the test body, the small one, is not at all interested in the massive body.
01:16It has a trajectory determined only by the space-time metric in the region where it is located.
01:21Now let's replace the massive body with another one.
01:25If before the test body, the small one, had a certain trajectory,
01:29now it will change its trajectory because it is attracted by another massive body.
01:34Classically, we could say that the forces of attraction have changed.
01:37In Einstein's thinking, however, we don't have forces and we are forced to admit that the metric of space has
01:43changed.
01:45Time curve around the small body.
01:47Once this small body has a different trajectory, because only the space-time metric determines the motion of the test
01:53body,
01:54the small one, as we saw last time.
01:56Who changed this space-time metric around the test body, the small one, once I brought another massive body nearby?
02:04The answer can only be the following, the massive body itself.
02:09What do we deduce from this, that the space-time metric around the small test body,
02:14the one that determines its motion, has been modified by the massive bodies.
02:18This figure shows an example.
02:21Here I've depicted the two bodies in space-time.
02:24The space-axis is horizontal and the time-axis is vertical.
02:28In Newtonian mechanics, at the top of the image, the massive body on the left attracts the test body on
02:34the right.
02:35In the theory of general relativity, the massive body deforms space-time everywhere,
02:39including at the events where the test body is located.
02:42The test body will follow a time-like geodesic in this now curved space-time and that's why it will
02:50move.
02:50It should be noted that not only space, but the entire space-time is curved.
02:56Of course, however, saying that these massive bodies are the ones that determine the metric of curved space-time is
03:02no longer enough.
03:03Unlike many of the assumptions made so far, this time Einstein had to pick up a pencil and actually calculate
03:10the equation.
03:11It tells us how the metric G of curved space-time is created by massive bodies for any coordinate system,
03:17T, X, Y, Z, even when chosen arbitrarily.
03:22What did he find? Who curves space-time?
03:24We might think it's the masses of massive bodies.
03:27After all, in Newtonian mechanics, they are the ones that determine the forces of attraction.
03:33However, Einstein was fully aware of the equivalence between mass and energy,
03:37which we also discussed in the context of special relativity.
03:41If they are equivalent, then we can just as well use energy to determine the metric tensor G,
03:47that is, the curvature of space-time.
03:49In the end, Einstein found that the most suitable description is given by the energy-momentum tensor,
03:54whose form you can see here for stellar dust moving non-relativistically.
03:59It is a square matrix, described in the four components of space-time,
04:03the first being time and the last three being space.
04:06In front of the matrix, we recognize the mass density rho.
04:10Then, on its diagonal at the top, we have the predominant element, the energy density.
04:17The other three diagonal elements give the densities associated with momentum,
04:22and the rest of the elements are associated with a transfer of momentum and energy.
04:27Of course, like all formulations, this energy-momentum tensor matrix
04:31takes different values at different events in space-time.
04:34In the end, Einstein found the correct form of the equations.
04:37In them appears not only energy, as expected,
04:39but a much more complex quantity called the energy-momentum tensor.
04:44The new function is actually a generalization of energy and momentum,
04:48being described at any point in space by the energy density, momentum density,
04:53and the transfer of momentum along various axes.
04:56I have to say that the most beautiful description of the properties of the energy-momentum tensor
05:01I found in Landau and Lifshitz's book,
05:03The Classical Theory of Fields.
05:04I recommend it to you as well.
05:08The energy-momentum tensor of matter is then placed in the so-called Einstein equation,
05:13which shows us how energy curves spacetime at each of its events.
05:18The equation looks like this.
05:19On the right side, we recognize the energy-momentum tensor of matter, T mu,
05:23together with Newton's constant, G, and the speed of light, C.
05:28The left side of the equation contains elements related only to spacetime curvature,
05:32now represented by the metric G mu.
05:35The indices are for spacetime.
05:37The other two elements come from this metric.
05:41The first is the Ricci tensor, denoted as R mu.
05:45The second is the scalar curvature of spacetime, denoted by the letter R.
05:50For completeness, these two elements are calculated from the metric
05:54using the Christoffel symbol gamma, whose formula I showed in the previous episode.
05:59So what does Einstein's equation, which I'll show here once again, tell us?
06:03That the metric of spacetime, meaning its curvature,
06:06is generated by the energy-momentum tensor of the matter that exists in that spacetime.
06:12He was very excited about the equation he discovered, because it has a covariant form.
06:19Simply put, the descriptions are made using tensors, which, besides their matrix form,
06:25have well-defined transformation rules from one coordinate system to another.
06:30Einstein's equation, the most important one in the theory of general relativity,
06:35confirms that the laws of physics are the same no matter which arbitrary coordinate system we choose.
06:40Einstein was very excited about this aspect.
06:43That's why the new theory is called the theory of general relativity,
06:47as opposed to the theory of special relativity,
06:50which applies only to inertial reference systems.
06:57As I was saying, it took Einstein almost ten years of intense effort
07:01to discover the equation that bears his name.
07:04However, there was someone else who found the same equation at the same time,
07:08but it took him less than a year.
07:10We're talking about David Hilbert.
07:12He was one of the greatest mathematicians of the last two centuries,
07:15with important contributions,
07:18not only in mathematics, but also in physics.
07:21He lived in the same period as Einstein,
07:24back when the latter was searching for the equations that now bear his name.
07:28Drawn to the problem, even inviting Einstein to Göttingen
07:31to hold a few seminars on the subject,
07:33Hilbert chose a different approach,
07:35one in which not the energy-momentum tensor plays a decisive role,
07:39but rather the principle of least action.
07:42In Hilbert's view, as a mathematician,
07:45the solution to the problem of the space-time metric
07:48must be geometric, axiomatic, and general.
07:52After less than a year of searching,
07:54he found the correct equation,
07:55reportedly just a few days before Einstein,
07:58who had already been searching for it for many years.
08:00Although Hilbert was faster,
08:02he never claimed any priority over these equations
08:05because, as he himself admitted,
08:07he built his solution on the foundations already formulated by Einstein.
08:12However, Hilbert realized the importance of mathematics
08:15in the new physics that was developing,
08:16where mathematics was becoming more and more complex.
08:19There's a joke that he remarked,
08:21physics has become too difficult
08:23to be left in the hands of physicists.
08:26And that was today's lesson.
08:28If you want me to resume the physics lessons,
08:30write to me in the comments.
08:31You can find the old ones in the physics lesson playlist.
08:35Until next time, please subscribe if you haven't already
08:38by pressing this subscribe button right now.
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08:44and here become channel members.
08:46I am Christian Presura.
08:48You are cool and I wish you all the best.
08:50Goodbye.
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