5.6 A Bit About Tensors (General Relativity)
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This article is from the Relativity and FTL Travel FAQ, by Jason W. Hinson jason@physicsguy.com with numerous contributions by others.
5.6 A Bit About Tensors (General Relativity)
In this section I will introduce just a few basic ideas which will give
the reader a feeling for what tensors are. This is simply meant to provide a
minimum amount of information to those who do not know about tensors.
Basically, a tensor is a geometrical entity which is identified by its
various components. To give a solid example, I note that a vector is a type
of tensor. In an x-y coordinate system, a vector has one component which
points in the x direction (its x component) and another component which
points in the y direction (its y component). If you consider a vector
defined in three dimensional space, then it will also have a z component as
well. Similarly a tensor in general is defined in a particular space which
has some number of dimensions. The number of dimensions of the space is also
called the number of dimensions of the tensor. Note that vectors have a
component for each individual (one) dimension, and they are called tensors
of rank 1. For other tensors, you have to use two of the dimensions in order
to specify one component of the tensor. In x-y space, such a tensor would
have an xx component, an xy component, a yx component, and a yy component.
In three-space, it would also have components for xz, zx, yz, zy, and zz.
Since you have to specify two of the dimensions for each component of such a
tensor, it is called a tensor of rank 2. Similarly, you can have third rank
tensors (which have components for xxx, xxy, ...), fourth rank tensors, and
so on.
So that you aren't confused, I want to explicitly note that the
dimensionality of a tensor (the number of dimensions of the space in which
the tensor is defined) is independent of the rank of the tensor (the amount
of those dimensions that have to be used to specify each component of the
tensor). In any dimensional space, we can have a tensor of rank 0 (just a
number by itself, because it is not associated in any way with any of the
dimensions), a tensor of rank 1 (like a vector--it has a component for every
one dimension you can specify), a tensor of rank 2 (it has a component for
every pair of dimensions you can specify), etc.
Now we look at a very important property of tensors. In fact, it is the
property which really defines whether a set of components make up a tensor.
This property involves the question of how the tensor's components change
when you change the coordinate system you are using for the space in which
the tensor is defined. So, let's consider an example in two dimensional
space where you go from some coordinate system (call the coordinates x and
y) to some other coordinate system (call these coordinates x' and y'). There
will be some sort of relationship between the two systems. For example, say
we start at some point in this space such that our coordinates are (x,y) and
(x',y') (depending on which coordinate system you are using). Now, say we
move an "infinitesimal distance" in x (using the first coordinate system).
Call that distance dx. When we do so, we may have changed our x' position
(using the second coordinate system) by some infinitesimal amount, dx'.
Also, we may have changed our y' position by some amount dy'. We can use
these concepts of infinitesimal changes to define some relationships between
the two systems. We can answer the question "how does x' change when x
changes at this point" by noting the ratio, dx'/dx. Similarly we can write
dx/dx' to denote how much x changes with changes in x' at some point, and
dy'/dx denotes how y' changes with changes in x.
Please understand that these are not simply ratios of definite numbers.
For example, dx'/dx is not necessarily the inverse of dx/dx' because dx in
one expression is NOT the same as dx in the other. The first expression uses
dx in the following context: "If I hold y constant and change x by an amount
dx, x' and y' might change by amounts dx' and dy'. Take the amount that x'
changes (dx') and divide it by the amount I changed x (dx)." The second
expression uses dx in the following context: "If I hold y' constant and
change x' by an amount dx', x and y might change by amounts dx and dy. Take
the amount that x changes (dx) and divide it by the amount I changed x'
(dx')." You can see that the dx in the former context does not have to be
the same amount as dx in the latter. So, when I write dx'/dx or dx/dx' or
dy/dx' etc, you must understand that the form of these ratios (what's on top
and what's on bottom) defines how they are produced, and they are not just
ratios of definite numbers. (Those who know something of calculus will
obviously recognize these terms as simple partial derivatives, but
anyway....)
Now, all together there are four of these ratios which denote how the
x' and y' coordinates change with changes in x and y:
dx'/dx, dx'/dy, dy'/dx, and dy'/dy.
Similarly, there are four more to denote how x and y change with changes in
x' and y':
dx/dx', dx/dy', dy/dx', and dy/dy'.
In general the values of these ratios can depend on where you are on a
manifold, so each ratio is generally a function of x and y (or x' and y', if
you like).
Now, we have these ratios which help us relate one coordinate system to
another. If we have a tensor defined in this space, then we must be able to
use those ratios to find out how the tensor's components themselves change
when we go from considering them in one coordinate system to considering
them in the other. Let's consider a tensor of rank 1 (a vector) in a two
dimensional space. Let the vector, call it V, have an x component (V_x) and
a y component (V_y). Then, the rules for finding the x' and y' components of
the vector at some point are the following:
(Eq 5:4)
V_x' = dx'/dx V_x + dx'/dy V_y
and
V_y' = dy'/dx V_x + dy'/dy V_y.
That is the way in which this type of first rank tensor must transform
from one coordinate system to another. Note that we can write both equations
in Equation 5:4 by using the following:
(Eq 5:5)
V_a = SUM(b = x,y) [da'/db V_b]
In that expression, "a" can be either x or y (so we actually have two
equations--those in Equation 5:4). Also, the right side of the equation is a
summation where the first term in the summation is found by letting b = x,
and the second term is found by letting b = y. Further, we could make this
expression more general by noting that it will be true for a space with
higher dimensions when we let "a" be any one of those dimensions and let the
sum with b extend over all the dimensions.
The fact that the physical components of a vector do actually transform
this way is what makes the vector a tensor. However, we should note that not
all types of vectors transform this way.
To show this is so, first we will consider a function which has a value
at every point in x-y space. Call the function f(x,y). Such a function is a
0 rank tensor, because at any point in the space, it has some single,
numerical value (it does not have components for x and y like a vector
does--you can't ask "what's its value in the x direction", or "what's its
value in the y direction", because it has only a single number at any
point). Note that if we change to another coordinate system, the value of f
at some physical point in the space will not change. Because it has no x or
y component, it is invariant when you change coordinate systems, as are all
0 rank tensors. This is the way all 0 rank tensors must transform when you
change coordinate systems--they must be invariant.
Now, back to the point that there are other types of vectors which do
not transform as discussed earlier. Let's take the function we were just
discussing, f(x,y), at some point and ask "how does it change with small
changes in x?" If the function changes by an amount df when we move to
another x location a distance dx away, then we can write the expression
df/dx to tell how f changes with x. We can do the same in y and have the
expression df/dy. Then we could define a vector (call it G) which has an x
component (G_x) equal to df/dx at every point in x and y, while it has a y
component (G_y) equal to df/dy at every point. Now, what if we do this same
procedure in the x'-y' coordinate system. First, we need to convert f into a
function f'. We do this such that if a point in our space has coordinates
(x,y) in one coordinate system while the same physical point has coordiantes
(x',y') in the other coordinate system, then we want f(x,y) = f'(x',y').
That way f' is the proper representation of f in the primed coordinate
system. Now we again find a vector, G, and we will end up with the x' and y'
components of the G vector such that G_x' = df'/dx' and G_y' = df'/dy'.
We now want to figure out how to transform G from one frame to another.
First, we will look at G_x' = df'/dx' which says that G_x' comes from
knowing how f' changes with respect to x' (i.e. df'/dx'). To transform this
component of G, we must know how to find df'/dx' using G_x and G_y. This
means we will be using information about how f changes with respect to x and
y (i.e., using df/dx and df/dy). We will also need to use information about
how x and y change with respect to x'. Without taking the time to fully
explain the calculus involved, perhaps the following equation will not be
too surprising:
(Eq 5:6) df' df' dx df' dy -- = -- * -- + -- * -- dx' dx dx' dy dx'
Conceptually (though mathematicians would cringe a bit at this
explanation) one can imagine canceling out the dx in df'/dx * dx/dx' and
canceling out the dy in df'/dy * dy/dx' to see that in both parts of that
equation we are looking at information about df'/dx'. In the first case, we
are looking at how f' changes with respect to x' by way of how x changes
with respect to x', while in the second case we are looking at how f'
changes with respect to x' by way of how y changes with respect to x'.
Adding these two components together as we do in the above equation gives us
a full picture of how f' changes with respect to x' given information about
how f' changes with respect to x and y.
We further note that f' and f are actually the same physical function,
we just use the prime to indicate which coordinate system we are primarily
thinking of. Thus f and f' will both change in the same way with respect to
changes in x and y (i.e. df'/dx = df/dx and df'/dy = df/dy. We therefore
rewrite Equation 5:6 as
(Eq 5:7)
df' df dx df dy
-- = -- * -- + -- * --
dx' dx dx' dy dx'
dx dy
= G_x* -- + G_y* --
dx' dx'
Note that we have substituted G_x = df/dx and G_y = df/dy. The above
equation provides the transformation of G_x' given the components of G in
the (x,y) coordinate system. Similarly, we can also find the transformation
of G_y'. In the end, simply because of the way this vector is defined, it
transforms as follows:
(Eq 5:8) G_x' = dx/dx' G_x + dy/dx' G_y and G_y' = dx/dy' G_x + dy/dy' G_y
As before, we can rewrite these two equations as follows:
(Eq 5:9) G_a' = SUM(b = x, y) [db/da' G_b]
Note that we are using ratios like db/da' rather than da'/db (which we used
earlier). That means that this is a different type of vector (because it
transforms in a different way). The vector we discussed earlier (V) is
called a contravariant vector, and the fact that it transforms as shown in
Equation 5:5 is what defines it as that type of vector. The G vector is
called a covariant vector, and it is defined as such because it transforms
as shown in Equation 5:9. Usually, we express which type of vector we have
by the way we denote its components. For contravariant vectors, we denote
their components by putting their indexes (the x or the y) in superscripts:
x y
V and V (or V^{x} and V^{y}),
While we denote the components of covariant vectors by putting their indices
in subscripts:
G and G (or G_x and G_y) x y
With this notation, the two different transformations begin to take on
an easy to remember form. See if you can figure out how the "upper" indices
and the "lower" indices match up on both sides of the two transformation
equations when they are written as follows:
(Eq 5:10)
a' da' b
V = SUM(b = x,y) -- V
db
and
(Eq 5:11)
db
G = SUM(b = x,y) -- G
a' da' b
Notice that the superscript (or subscript) on one side remains "upper" (or
"lower") in the ratio on the other side. Also, note that the summation is
always over the index which is repeated on the right side, once in an
"upper" position and once in a "lower" position. This basic "formula" helps
to produce equations for all transformation in tensor analyses (note this in
the next part of this section).
It is interesting to note that in the normal spatial coordinates we are
used to using (Cartesian coordinates), db/da' = da'/db, and there is no
distinction between covariant and contravariant vectors. However, in other
systems, the difference is there and must be considered.
Further, we note that with higher rank tensors, they are also defined
by the way they transform from one coordinate system to another. For
example, consider a second rank tensor, U. It could be that both of its
indices are associated with the contravariant type of transformation (note:
the following actually denotes four equations because a'b' can be set to
x'x', x'y', y'x', or y'y'):
(Eq 5:12)
a'b' da' db' xx da' db' xy da' db' yx da' db' yy
U = -- * -- U + -- * -- U + -- * -- U + -- * -- U
dx dx dx dy dy dx dy dy
[ da' db' ce ]
= SUM(c & e vary over all dimensions) [ -- * -- U ]
[ dc de ]
Or they could both be associated with the covariant type of transformation:
(Eq 5:13)
[ dc de ]
U = SUM(c,e) [ -- * -- U ]
a'b' [ da' db' ce ]
Or it could be a mix of the two:
(Eq 5:14) a' [ da' de c ] U = SUM(c,e) [ -- * -- U ] b' [ dc db' e ]
Finally, we will see in the next section that any contravariant tensor
also has a covariant form (and vice-versa), and we can transform from one
form to the other if we know the geometry of the manifold on which the
tensors are defined.
And that about ends our introduction to tensors. To sum up, they are
geometric entities which have components denoted by some number of indices.
Each index can be any of the dimensions in which the tensor is defined, and
the number of indices needed to specify a component of a tensor is called
the tensor's rank. We are familiar with 0 and 1 rank tensors (numbers--or
"scalars"--and vectors). Finally, the way one transforms a tensor from one
coordinate system to another depends on the type of tensor, and it (in fact)
defines what it actually is to be a tensor. Each index of a tensor will
transform in either a contravariant way or a covariant way.
These are the basic ideas behind tensors, and they allow us to define
some very powerful mathematics. If you are familiar with the usefulness of
vectors, then you have touched the surface of the usefulness of tensors in
general. In the following section, we will look at two particular tensors,
and we will see that they can be quite useful.
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