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Tensors
As near as I can figure, these are three plus dimensional matrices that are used in high level physics. Does anyone know exactly how these work, in particular as they relate to relativity?
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I haven't studied relativity, but I do have some knowledge of tensors in other areas, such as stress.
I'm not going to go into the specific mathematics, because of the amount of prerequisite knowledge about advanced geometry and calculus. If you have studied math or physics at graduate levels this book is very good. If you're starting with undergrad / college math skills, aren't a scaredy pants and have time to read ~250 pages (the book is 1000 but they get to Tensors by 250) the this book is excellent.
So now that's out of the way, let me try and give a "hands waving" simple picture of what Tensors do. When you have 3-dimensional space then any point can be identified with three numbers (x, y, and z-coordinate). These numbers are lumped together to form a vector. Similarly, at any given point you can have a pointer in some direction (e.g. an electric field) which is also described by a vector. Now, in order for you to be able to do anything with those things you need rules for adding and multiplying them. Vector multiplication, as it is taught in high school is just multiplying the elements at the same location in the vector and then adding the result:
A.X = (a,b,c).(x,y,z) = a*x + b*y + c*zWhat they don't teach you is that that works only in the very special case of flat space. If you use the above formula to calculate the area of a triangle (which can be defined with two vectors) then you get the wrong result if you happen to be on a sphere (e.g. think about the triangle that has all three sides on the equator). So the correct way to do the above calculation is to use a metric g (which is a matrix) thuslyA.X = A.g.X = (a,b,c).((g11,g12,g13),(g21,g22,g23),(g31,g32,g33)).(x,y,z)In flat space g is diagonal (only g11, g22, and g33 nonzero, in fact g11=g22=g33=1) and you get back the "easy" formula. The metric tells you how to modify A.X depending on what kind of space A and X are in. Essentially the metric contains all the information about the space you are in.= a*g11*x + a*g12*y + a*g13*z + b*g21*x*...
Notice that I haven't used the word tensor yet. What happens now is that I ask you, what happens to the calculation if I stretch and rotate the underlying space? Clearly, if this is just a change of using kilometers instead of inches and / or turning the map upside down, this should not affect the end results of any calculation. A tensor is an object that allows for, and obeys, certain transformation rules such that this invariance is the case! You can see how this is important for physics, as the result of our calculations shouldn't depend on the units we use or the coordinate system.
The metric is a tensor of rank 2 (essentially matrix). If you look at the math of A.g.X you see that it has to be a matrix since it combines and "mixes up" two vectors (which are rank 1 tensors). This can be generalized to e.g. multiplying a vector with a matrix, in which case we need a rank 3 tensor, multiplying two matrices -> rank 4 tensor. You may be familiar with how these all look like in flat space.
So far so good, I think that covers what tensors are. Now to the "how do they apply to general relativity", i.e. gravity. Think about the metric and how it contains the information about the "form" of space. The next step is to realize that the metric can change from one point to another. Just like coordinates are different form one point to another (well duh!), or e.g. an electric field varies from point to point. What Einstein realized was that matter (and energy) distort space and that these distortions can be thought of as variations in the metric of space!
To do the math that this idea entails you need to, at the very least, somehow relate how the metric changes from one point to the next, i.e. relate two metrics, so you need a rank 4 tensor. This is the Riemann curvature tensor, effectively this is a 4 dimensional "matrix" R_a_b_c_d (this is a very, very hand-wavy explanation of this truly magnificent and awesome entity). A very important feature is that since R_a_b_c_d relates metrics, and relations are themselves made with metrics, R_a_b_c_d is entirely defined by the metric of space, nothing else goes into it.
If you look at the equation of general relativity:
There is T_a_b which describes the distribution of energy and matter in space, g_a_b is the metric, R_a_b is called the Ricci tensor and it is obtained from the Riemann curvature tensor by taking a trace once (remember how a trace over a matrix yields just a number reducing the rank by 2), and R is the curvature of space which you get from the Ricci tensor by contraction with the metric (and kappa is just a constant). Sometimes the right hand side includes a term Lambda g_a_b, with Lambda the cosmological constant. Simplifying units and notation, this can be written as
which is a truly remarkable equation. Just like E=mc^2, which iconically relates the equality of energy and mass, this equation has on its left hand side nothing but the metric and objects derived thereof and on the right hand side energy and mass. It effectively says "The geometry of space is dictated by the distribution of energy and mass and vice versa".