12.5 General Relativity

Key Takeaways

welcome back to 820 special relativity so in our quest to understand how we get to general relativity there's two things to consider the first one this lecture is not meant to give you a full description of general relativity but just a view into where this might lead where this discussion might lead so in this quest we can understand the theory of general relativity as a theory on how to patch together the different reference frames which are each can be described in special relativity so in the framework we discussed up to now and is valid in short intervals in space time consequence of general relativity are that space time is curved so we have modified geometries um and we learned that because of gravitational effects matter curves space-time as a consequence of that you know there must be modification of of gravity based on meta distributions and so there must also be gravitational waves gravitational lenses which bend light black holes and there's cosmological predictions coming out of general relativity so let's have a discussion first what does it mean to have a changed or modified geometry what could that mean so you're all used to euclidean geometry where when you draw a triangle you add up all the angles to 180 degrees if you draw two parallel lines they never cross they also don't diverge but if you have a modified geometry for example the geometry on the sphere like on our globe the angles do not add up to 180 degrees they actually the sum is larger than 180 degrees and parallel lines will cross we would call this kind of space positively curved but you can have the opposite example like on a saddle so you can have other spaces and other curved spaces and they can be negatively skirved in this example if you add up all angles you find they add up to less than 180 degrees parallel lines will not cross but they will diverge okay so mass has changed the geometry of spacecraft time you know we just talked about light bending and because of the change in geometry light will not go on a straight line anymore but will bend around massive object space time is curved geometry of space-time tells us how the masses move you can think about um a trampoline when you put a heavy object on a trampoline all the other objects on the trampoline will gravitate towards the heavier object and that's kind of a picture on how spacecraft spacetime actually looks like einstein used those finding in order to redefine newton's first law and found the so-called einstein field equation so there's on one side of the equation there's a description of space time and its curvature and on the other side of the equation is the energy momentum momentum tensor the description and how energy energy and momentum of object is distributed and those two things space time and energy and momentum they're kind of interlinked in this in this equation so if you read this description you can read it from one side to the next space time tells matter how to move or you read it from the other direction say meta tell space crime space time how to curve there is an equation and you can just read it from the left to the right or from the right to the left so our understanding here it says space and time are not fixed things you know [Music] matter through which matter and energy moves through the matter and energy then itself defines space-time and matter because of that time sp space-time is dynamical it's changing it's interacting with the matter and with the energy this is a super exciting picture from hubble the hubble space telescope and you see galaxies um but what you also see is those structures which looked like something the light has gone through lenses those lenses are actually meta distributions galaxies themselves which actually lead to the bending of the light and those lensing effects okay if you want to summarize um general relativity you can first say that space time is curved and it follows the pseudo romanian monifold with a specific metric we have seen the metric before with minus plus plus plus and the relationship between matter and curvature is given by the einstein equation and here i give you a slightly different form where there is the dynamics again on one side and the energy momentum on the other side let's just look at one example here so we discussed in special relativity invariant intervals right and we had this delta s squared or we had different name for i given by minus dt square plus dx square plus dy square plus dz squared we could have just written this in polar coordinates as well where you find the d r square and r square d theta square and then r square sine square theta d phi squared okay same thing it's just a different coordinate system so as a solution to einstein equation we find something which looks very very similar that's not a surprise as we find general relativity as a patchwork of small spaces of general special relativity so the solutions might be very similar okay in the solution found here by the so-called schwarzschild solution which are unique solution vacuum with spherical symmetry of a meta distribution so you have a spherical meta distribution like our sun right and this is solution which describes um space time around this you find this invariant interval here has two interesting features there's two singularities in here so you find this should be a minus one you find those two singularities one is at r equals zero that's kind of expected in the middle of the mass distribution this thing is not defined anymore there's no mass left but there's also a second singularity at two gm this is called the so-called schwarzschild radius and it defines if you get to the singularity you basically don't define anymore this invariant interval you can think about the black hole the surface of a black hole as this singularity at this r values at those singularities everything becomes timeline time like or everything within the radius becomes time-like you