The Meaning of the Metric Tensor

possibly the concept most crucial to general relativity is that of the metric tensor the 10 numbers which assigned to every point of space and time rule our modern physics but despite the importance of this concept it's hard to find an explanation of it beyond just merely superficial mathematical definitions is there a better way we can understand its significance [Music] and what does it ultimately reveal about the nature of our reality this is dialect and today we begin exploring the meaning of the metric tensor general relativity is so buried under mind-bending ideas and esoteric mathematics that the basic secret of the theory might easily escape your notice all you're doing in general relativity is making a map a map of space and time that's it you are literally a space-time cartographer when european seafarers and navigators realized that the world was round they needed a way to depict a spherical surface on a flat piece of paper that they could hold in their hands that is they had to take a two-dimensional curved surface and map it onto a two-dimensional flat surface similarly when physicists discovered our four dimensions of reality were not flat but formed a single curved totality they needed a way to depict a four-dimensional curved surface using only the resources allotted to a casual observer i.e the four independent measurements of space and time meaning they had to take a four-dimensional curved surface and map it onto a four-dimensional flat surface now over the centuries cartographers have developed many different types of map projections there's the mercator robinson goods the azimuthal equidistant etc all of these maps depict the curved totality of the earth in different ways and while no single map encompasses the entire earth accurately each offers its own advantages according to the various nautical geographical or political needs of its user similarly physicists have developed many different maps of space-time these are referred to as coordinate systems or coordinate charts we have schwarzschild coordinates kruskal chakras coordinates flrw coordinates etc just as with maps of the earth no single coordinate system can encompass the entirety of a space-time region while accurately preserving proper distances and proper times but each can highlight or emphasize particular qualitative features about that space-time region which may or may not be useful to the physicist now every good map needs a bar scale and as we discussed in our prior video a metric is simply just that a bar scale or the mathematical object which tells us how distances on our map translate to distances in the real world in the case of our space time map it tells us how coordinate distances and coordinate times translate to proper distances and proper times across different locations of our map now you're probably used to a map having only one bar scale but a map of something like the earth actually requires a bar scale at every single point since distances and areas everywhere are continuously being distorted on this robinson map for instance a centimeter interval across greenland does not represent the same distance as a centimeter interval across equatorial africa similarly on our map of spacetime we'll need a metric or bar scale at every single space-time point to indicate how our space-time distance is changing across the map this comparison is important so we're going to reiterate it on a regular map a metric takes us from physical distances on the map to physical distances on the earth at each and every point while on a space-time map the metric takes us from coordinate distances on the map to space-time distances on the space-time manifold at each and every point but what is space-time distance exactly well it's more properly termed the space-time interval and it corresponds to the amount of time that would elapse on a clock located at a particular space-time point as measured by an observer also located there so essentially our space-time metric tells us how fast or slow clocks all across the universe are ticking with respect to our map user's local reference clock but since the space-time interval contains information not just about how time is changing across the map but also about how space is changing too if you want to keep all this information separate you're gonna need more than just one number at each point on your map in fact you're gonna need 10 numbers at each point and this is where the tensor part comes in how exactly well in our last video we explored the basic topological procedure for going from a curved surface to a flat surface or vice versa basically you cut your surface or manifold into many small nearly flat pieces then you stretch and or skew each piece before rearranging all the pieces into a new surface because this process takes a distance element that is a right triangle and turns it into a distance element that is an acute or obtuse triangle your distance goes from being described by the pythagorean theorem to being described by the law of cosines or one of its higher dimensional analogs the components of the metric tensor are then just the various lengths and cosines of angles needed to determine the length of your new distance element as an example of this overall process let's take a look at an equirectangular projection of the earth this projection takes the earth's lines of latitude and longitude and gives them equal spacing on a flat map thus forming a standard cartesian coordinate grid let's designate our axes by the variables x and y the values of y range from 0 to pi and the values of x range from 0 to 2 pi okay now that we have our flat surface let's follow the basic topological procedure for transforming it into a curved surface first let's cut the map into many tiny squares next let's impose the following metric on each square this metric or more precisely the metric tensor which we are assigning to each tiny square piece has three numbers or components the first component tells us by what amount to stretch or shrink the x length of the piece the value of this component is the sine of the value corresponding to the y-coordinate of that piece so this tells us that the x-length of pieces near the equator where the sine of the y-coordinate value is equal to nearly 1 will not be stretched or shrunk whereas the x-length of pieces near the poles or the sine of the y-coordinate is equal to nearly zero will be shrunk to close to zero meanwhile the x lengths of all the pieces in between will be shrunk an intermediate amount now the second component tells us how much to stretch or shrink the y length of the piece here everywhere on the map that component is equal to one so this tells us that the y length of all our pieces remains unchanged finally our third number or component is zero this tells us that the cosine of the new angle between our transformed axes should be zero the cosine of 90 degrees is zero and since the old angle between the axes of our pieces was already 90 degrees this means there will be no skewing of any of our pieces across the map we see we now have all the correctly sized pieces of a new surface with a little more information about the coordinate transformation we can rearrange our pieces into a new surface [Music] notice that there are some gaps between pieces as well as some places where the pieces overlap and are touching which we see especially towards the poles this is because we did not cut our map into small enough pieces but the theory of differential geometry assures us that if our pieces were infinitesimally sized the surface of the sphere will approach its proper shape and area now rather than an equirectangular projection if you instead started with a projection like the robinson one where the lines of latitude and longitude didn't intersect at right angles you would end up with the third component in your metric tensor being non-zero this would indicate that in addition to needing to be stretched or shrunk pieces across the map would need to also be skewed before being rearranged into their new surface zooming into this spot in north america for instance we see that the angle between the lines of latitude and longitude here is about 45 degrees the tiny square patch which resides there then would have to be skewed until the proper angle of 90 degrees between the lines of longitude and latitude was reached thus the original coordinate angle between the x and y axes on your patch which was 90 degrees would get stretched to a value greater than 90 degrees and it's the cosine of that angle which essentially gives you the value of your third metric component with the caveat that it has to be multiplied by the magnitude of the other two components first all right we've mastered the two-dimensional metric so let's take it to the next level and look at a three-dimensional metric now as with our two-dimensional case we're going to start out with a flat surface for a map which in three dimensions is just your conventional euclidean 3d space previously we cut our map into many tiny squares now we'll cut our map into many tiny infinitesimally sized cubes dx by dy by dz then we'll assign a metric tensor to each infinitesimal cube on the map okay let's zoom in on a particular cube in three dimensions the metric tensor has six components three for the three sides of the cube plus another three for the three angles between pairs of axes these components at any point will be a function of the map coordinates since again our bar scale is changing across different points of the map so let's just say that those functions at this specific point where our cube of interest is located end up giving us something like these numbers now the first instruction from the metric tensor at this point is to multiply dx by 1.5 the second instruction is to multiply d y by three-fifths the third instruction is to multiply dz by four-fifths next the fourth instruction is to take the angle between the x and y axes and shrink it until it equals 70 degrees similarly the fifth and sixth instructions tell us to make the angle between the x and z axes equal to eighty degrees and the angle between the y and z axes equal to 60 degrees we now have the correctly sized piece that belongs to our new curved manifold moving to the next point in our three-dimensional map we read the new metric tensor instructions there to size that piece as well repeating this for every point in our 3d map we arrive at a collection of properly sized pieces for our new manifold now were we in possession of a higher dimensional coordinate transformation we could conjoin these pieces into a curved entirety like we did with the earth [Music] however there's a problem with this the problem isn't that we can't imagine more than three dimensions rather it's that we may simply not be in possession of a higher dimensional coordinate transformation to begin with and backwards deducing one may not be possible or physically meaningful at any rate if our map is equipped with a metric tensor field we will at least have an understanding of how the manifold looks locally at every point well that's it you're just steps away from taking the metric tensor into the four dimensions of space-time and beginning to really understand how general relativity works but we're out of time for today so we'll continue our exploration of the meaning of the metric tensor in a future video so stay tuned this has been dialect thanks for watching you

Channel: Dialect

Views: 160,684

Keywords: general relativity, physics, topology, metric tensor, tensors, math, learn

Id: Dn0ZZRVuJcU

Length: 19min 22sec (1162 seconds)

Published: Sat May 07 2022