How Mass WARPS SpaceTime: Einstein's Field Equations in Gen. Relativity | Physics for Beginners

hey what's up you lot path here and in today's video we are going to be discussing the basics of einstein's field equations as always you don't need to know any advanced mathematics for this hopefully we'll be understanding these terms in a fairly intuitive way if you enjoyed this video then please do hit the thumbs up button and subscribe to my channel for more fun physics content hit the bell button to to be notified whenever i upload and check out my patreon as well if you'd like to support me on there okay let's get into it the first thing that we need to understand is that the einstein field equations are essentially the governing equations in the theory of general relativity they're basically the big boy equations that relate some of the most important concepts in the theory of gr now notice how i keep saying equations rather than equation well that's because what we see here is a rather convenient way of writing multiple equations these little subscripts mu and nu can take different numerical values and for each value of mu and nu we have a particular equation now we'll understand the specifics of mu and nu as well as the exact values that they can take very shortly but basically this way of writing einstein's field equations is just a convenient way of writing multiple equations now before we take a detailed look at what each one of these terms actually means another thing that we need to know is that these equations are dealing with tensors g subscript mu nu is a tensor lowercase g subscript mu is a tensor and t subscript mu is a tensor as well tensors are really interesting mathematical objects and we're not going to go too deeply into them in this video instead for our purposes we will be treating them simply as matrices now for those of you that are familiar with matrices you can skip to this timestamp here if you don't want to be bored by my rather basic description of them but if you're not familiar with matrices stick around now matrices basically look something like this they essentially contain lots of little bits of information with each bit of information referring to something very specific unlike this explanation i guess so let's take an example we could create a matrix for the following scenario let's say we've got five bags of apples each bag contains three apples one red one yellow and one green we label each bag with a number one two three four and five and within each bag we label the apples as well all the red apples are called apple number one the yellows are apple number two and the greens are apple number three we then find the masses of each of those apples here's a table showing those masses we can actually represent all of this information in a matrix this is what that looks like notice that each row refers to a specific bag bag one back two back three back four or back five and each column refers to a specific number of apple or in this case specific color of apple and we can call this matrix m since it contains all the information about the masses of these apples well speaking generally we can represent a matrix like this using just the matrix elements each little piece of information in this matrix is known as a matrix element and if we had a generic matrix m we could represent it like this the element in the very top left position can be represented by m subscript one one because it's the element of matrix m that's found in row one column one basically referring to the bag labeled number one and the apple labeled number one in that bag we could also look at element m one two which refers to the first bag second apple or first row second column and so on and so forth the first number in the subscript refers to which row in our matrix we're looking at and the second number refers to which column and if we want to talk about a generic matrix element one of the matrix elements in this matrix we could call it m subscript alpha beta where alpha can take the values from one to five representing bags one to five or rows one to five and beta can be between one and three so apples one to three or columns one to three and so we see that matrices are basically just an interesting way of displaying information but they also have some very interesting mathematical properties when we start playing around with them in fact the tensors in einstein's field equations behave in a very similar way we can represent these tensors with matrices you can see that we're referring to specific matrix elements row mu column nu which by the way is also a really good way to see that these are referring to multiple equations let's say we've got mu is equal to one and nu is equal to one well then that's one equation another possible combination is mu is equal to one and u is equal to two or any other possible combination of mu and nu now interestingly the tense is referred to in einstein's field equations are represented by four by four matrices which means four rows and four columns now rather annoyingly but slightly usefully as well there's a convention that gets used where we label the rows and columns as rows 0 1 2 and 3 and columns 0 1 2 and 3 rather than 1 2 3 and 4. so we've got the zeroth row first row second row and third row and similarly we've got the zeroth column first column second column and third column in other words our subscripts mu and u can take the values of 0 1 2 and 3. again like i said for our purposes this is mathematical convention but it actually plays really nicely into some relativistic ideas specifically the subscript 0 1 2 and 3 refer to the four different dimensions used in the theories of relativity 0 refers to the time dimension and 1 2 and 3 refer to the spatial dimensions which we can call x y and z i'm paraphrasing of course things are a little bit more complicated but for now that's all we care about so now that we know that we're dealing with tensors represented by matrices specifically four by four matrices let's work out what each one of these tenses is actually representing what does it mean the first tensor we'll be looking at is t subscript mu nu it's known as the stress energy tensor essentially it contains information about the distribution of stuff in the region of spacetime that we happen to be considering stuff meaning mass and energy many of you might be familiar with the idea that mass and energy are equivalent exchangeable if you will according to einstein's famous equation e is equal to mc squared though of course this is a slightly simplified version of that equation but it still conveys the same idea and the stress energy tensor essentially contains information about how this matter and energy is distributed throughout the region of space-time that we happen to be thinking about in many cases it's actually the whole universe but that's cool too for example there's a term that looks at the energy density in that region how energy is distributed over a certain volume as well as momentum density terms and shear stress terms and pressure terms which all sound rather complicated but essentially just the distribution of stuff and energy and how it moves around and how it is existing in the region of space time that we happen to be considering i realize that it's quite a vague explanation but i think to do it justice i'd have to make a separate video on these tenses all of these terms mass distribution energy distribution momentum flux and so on and so forth are contained within the stress energy tensor and they're really important because they contributes directly to the warping of space-time you may have seen a common description of general relativity that goes along the lines of a massive object say the sun for example being placed into the fabric of space-time and how that massive object bends the fabric of space-time around it that's because it has mass or it has energy or it has momentum or it has some sort of pressure going on that's what directly causes the warping of space time and this in fact brings us to the other side of the equation if t mu nu the stress energy tensor contains information about how stuff is distributed throughout space time the tensor capital g mu nu known as the einstein tensor contains information about the curvature of that spacetime fabric in other words a very basic description of iron stand field equations is that they basically tell us how the distribution of stuff in space-time ends up warping that space-time and equivalently how the warped space-time causes the stuff inside it to behave and our einstein field equations tell us exactly how that happens how much energy or how much mass do you need in order to warp space time in a particular way and consequently how objects behave in that warped space time now capital g mu nu the einstein tensor is a rather complicated function of a few other tenses such as the richie tensor we won't look into that in too much detail here but basically it tells us something about how different the space time is to flat space time and it's a function of the metric tensor lowercase g mu nu which by the way we can also see in this term here it's a very important tensor in relativity the metric tensor is effectively a measure of the shape of space time as opposed to the curvature which is more measured by the uppercase g menu einstein tensor the metric tensor deserves some explanation and again it's a tricky concept to cover in a video that's not about the metric tensor but let's think about it like this let's imagine an ant can live on a two-dimensional surface a flat plane it can move forward and backward or left and right but it's not allowed to move up or down now we label two points on this flat surface points a and b the ant is at a and it wants to get to b the shortest route to get from a to b is simply a straight line this is a rather simplified description of flat space-time where there's no energy no matter no nothing to warp the space-time fabric around itself but let's say that the surface that the ante is now sitting on is somehow warped so that it's on the outside of a sphere the surface is now curved but the and still follows the same rules it can move in what it thinks is the forward and backward direction or the right and left direction or of course any combination of those two directions it cannot jump up off the surface or burrow down into the surface so when getting from point a to b what's the shortest route it's now a curved path in this example if the and wanted to burrow down into the surface in order to get from a to b quicker that's kind of the equivalent of us traveling through a wormhole in our space-time fabric but anyway the important thing the metric tensor just tells us the shape of essentially the surface the end happens to be sitting on it tells us if it's a flat surface or a curved surface and it tells us exactly what shape the surface basically has except in our case there's obviously a few more dimensions to be accounting for now if capital g mu nu the einstein tensor which deals with the curvature of space-time is dependent on the shape of space-time as we would expect then why is there a separate term in the einstein field equations with the g-mu nu in it well this is because it accounts for another phenomenon entirely this weird little constant here this lambda is known as the cosmological constant and we could go on and on and on about it currently in our mathematics it accounts for the fact that our universe is expanding at a faster and faster rate we've observed galaxies really really far away from us moving away from us faster than galaxies that are slightly nearer to us we've got lots of evidence that currently suggests that the universe is not only expanding getting bigger space-time is stretching but it seems to be stretching faster and faster this is not really the warping of space-time due to the existence of mass or energy within that space-time this is something else entirely this is inherent to the space-time so for this reason we currently think the cosmological constant has a positive value which suggests an expanding universe this constant has a rather strange history initially einstein published his equations without the cosmological constant term and then he chucked it in then he felt like it was his biggest blunder and then realized that actually it does need to be there based on experimental observable evidence again this deserves a video of its own and i'd like to make that soon so keep an eye out for that how could i pass up the opportunity to talk about something that einstein himself described as his own biggest blunder now these other constants you may be familiar with eight hopefully you're familiar with as well as pi and g which is the universal gravitational constant and c which is the speed of light these constants basically tell us exactly how much mass or how much energy or how much whatever in the stress energy tensor is needed in order to create a particular warping of space-time they can be thought of as the coupling constants they essentially tell us how strong the effect mass or energy has on the warping of space-time if this constant was bigger then we would see more warping of space-time for the same amount of mass if this constant was smaller we'd see less warping but in our universe the constant seems to be 8 pi g over c to the power of 4. so anyway we've basically looked at a simple description of einstein's field equations and what information they convey they essentially relate the shape and curvature of space-time to the distribution of mass and energy within it therefore they tell us exactly how mass or energy causes space-time to warp and consequently how this warping results in mass and energy behaving in that warped space-time it also accounts for the fact that the expansion of our universe seems to be accelerating using the lambda or cosmological constant term i haven't really shown you exactly how each one of these tenses relates to the physical ideas that we've discussed but again future video one thing i will say though is that finding solutions to these einstein field equations is very very difficult but we have found some one of them is a flat space time which tells us that a region of space time without any mass or any energy exists as flat space time just as we'd expect no warping no nothing another solution to these einstein field equations is given by the schwarzschild metric which describes stationary black holes and it describes how spacetime warps around these stationary black holes i've actually made two videos on stationary black holes check out my black holes playlist as well as one on the ker solution or the ker metric which talks about rotating black holes so please do check out those videos and in the meantime i'd like to thank you so much for all your support and thank you so much for watching this video if you enjoyed it please do leave a thumbs up and subscribe to my channel for more fun physics content hit that bell button if you want to be notified when i upload and please do check out my patreon if you'd like to support me on there once again thanks so much for watching and i will see you really soon [Music] you

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Length: 14min 15sec (855 seconds)

Published: Tue Dec 01 2020