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Intro to the metric tensor
With these two assumptions and this convenient property in hand, we will now examine what it means to say that spacetime is curved.
====Flatness versus curvature====
Let's start by considering the simplest possible geometry<ref>Well, the simplest possible ''interesting'' geometry, anyway.</ref>: the [[Euclidean plane]].
The Euclidean plane is an infinite, flat, two-dimensional surface. A sheet of paper is a good approximation of the Euclidean plane. Onto this plane, we can project a set of [[Cartesian coordinates]]. By "Cartesian," we mean that the coordinate axes are straight lines, that they are perpendicular, and that the unit lengths of the axes are equal. A fancier term for a Cartesian coordinate system is an ''orthonormal basis.''
Note carefully the distinction between the Euclidean plane and Cartesian coordinates. The plane exists as a thing in and of itself, just as a blank piece of paper does. It has certain properties, which we'll get into below. Those properties are ''intrinsic'' to the plane. That is, the properties don't have anything to do with the coordinates we project onto the plane. The plane is a geometric object, and the coordinates are the method by which we ''measure'' the plane. (The emphasis on the word ''measure'' there is not accidental; please keep this idea in the foreground of your mind as we continue.)
Cartesian coordinates are not the only coordinates we can use in the Euclidean plane. For example, instead of having axes that are perpendicular to each other, we could choose axes that are straight lines, but that meet at some non-perpendicular angle. These types of coordinates are called ''oblique.''
For that matter, we're not bound to use straight-line coordinates at all. We could instead choose [[polar coordinates]], wherein every point on the plane is described by a distance from a fixed but arbitrary point and an angle from a fixed but arbitrary direction. Polar coordinates are often more convenient than Cartesian coordinates. For example, when navigating a ship on the ocean, the location of a fixed point is usually described in terms of a ''bearing'' and a ''distance,'' where the distance is the straight-line distance from the ship to the point, and the bearing is the clockwise angle relative to the direction in which the ship is sailing. Polar coordinates in two and three dimensions are often used in physics for similar reasons.
But there's a fundamental problem with polar coordinates that is not present with Cartesian coordinates. In Cartesian coordinates, every point on the Euclidean plane is identified by ''exactly one'' set of real numbers: there is precisely one set of <math>x</math> and <math>y</math> coordinates for every point, and every point corresponds to precisely one set of coordinates.
This is not true in polar coordinates. What are the unique polar coordinates for the origin? The radial distance is obviously zero, but what is the angle? In actuality, if the radial distance is zero, ''any'' angle can be used, and the coordinates will identify the same point. The one-to-one correspondence between points in the plane and pairs of coordinates breaks down at the origin.
In mathematical terms, polar coordinates in the Euclidean plane have a ''coordinate singularity'' at the origin. A coordinate singularity is a point in space where ambiguities are introduced, not because of some intrinsic property of space, but because of the coordinate basis you chose.
So clearly there may exist a reason to choose one coordinate system over another when ''measuring'' — there's that word again — the Euclidean plane. Polar coordinates have a singularity at the origin — in this case, a point of undefined angle — while Cartesian coordinates have no such singularities anywhere. So there may be good reason to choose Cartesian coordinates over polar coordinates when measuring the Euclidean plane.
Fortunately, this is always possible. The Euclidean plane can ''always'' be measured by Cartesian coordinates; that is, coordinates wherein the axes are straight and perpendicular at their intersection, and where ''lines of constant coordinate'' — picture the grid on a sheet of graph paper — are always a constant distance apart no matter where you measure them.
Imagine taking a piece of graph paper, which is printed in a pattern that lets us easily visualize the Cartesian coordinate system, and rolling it into a cylinder. Do any creases appear in the paper? No, it remains smooth all over. Do the lines printed on the paper remain a constant distance apart everywhere? Yes, they do. In technical mathematical terms, then, the surface of a cylinder is ''flat.'' That is, it can be measured by an orthonormal basis, and there is everywhere a one-to-one correspondence between sets of coordinates and points on the surface. It's possible ''not'' to use an orthonormal basis to measure the surface; one might reasonably choose polar coordinates, or some other arbitrary coordinate system, if it's more convenient. But whichever basis is actually used, it's always ''possible'' to switch to an orthonormal basis instead.
Now imagine wrapping a sheet of graph paper around a basketball. Does the paper remain smooth? No, if we press it down, creases appear. Do the lines on the paper remain parallel? No, they have to bend in order to conform the paper to the shape of the ball. In the same technical mathematical terms, the surface of a sphere is ''not flat.'' It's ''curved.'' That is, it is not possible to measure the surface all over using an orthonormal basis.
But what if we focus our attention only on a part of the sphere? What if instead of measuring a basketball, we want to measure the whole Earth? The Earth is a sphere, and therefore its surface is curved and can't be measured all over with Cartesian coordinates. But if we look only at a small section of the surface — a square mile on a side, for instance — then we can project a set of Cartesian coordinates that work just fine. If we choose our region of interest to be sufficiently small, then Cartesian coordinates will fit on the surface to within the limits of our ability to measure the difference.
The surface of a sphere, then, is ''globally curved,'' but ''locally flat.'' In physicist jargon, the surface of a sphere can be ''flattened'' over a sufficiently small region. Not the whole sphere all at once, nor half of it, nor a quarter of it. But a sufficiently small region can be dealt with as if it were a Euclidean plane.
But this brings up an important point. The ''entire'' surface of the sphere is curved, and thus can't be approximated with Cartesian coordinates. But a sufficiently ''small'' patch of the surface can be approximated with Cartesian coordinates. This implies, then, that "curvedness" isn't an either-or property. Somewhere between the locally flat region of the surface and the entire surface, the ''amount'' of curvature goes from none to some value. Curvature, then, must be something we can measure.
====The metric tensor====
It is a fundamental property of the Euclidean plane that, when Cartesian coordinates are used, the distance <math>s</math> between any two points <math>A</math> and <math>B</math> is given by the following equation:
:<math>
s^2 = \Delta x^2 + \Delta y^2\,
</math>
where <math>\Delta x</math> and <math>\Delta y</math> are the distance between <math>A</math> and <math>B</math> in the <math>x</math> and <math>y</math> directions, respectively. This is essentially a restatement of the universally known [[Pythagorean theorem]], and in the context of general relativity, it is called the ''metric equation.'' Metric, of course, comes from the same linguistic root as the word ''measure,'' and since this is the equation we use to ''measure'' distances, it makes sense to call it the ''metric'' equation.
But this ''particular'' metric equation ''only'' works on the Euclidean plane with Cartesian coordinates. If we use polar coordinates, this equation won't work.<ref>This is trivial to demonstrate. If <math>r</math> is zero and <math>\theta</math> is non-zero, then <math>r^2 + \theta^2</math> will be non-zero for a vector that obviously has no length.</ref> If we're on a curved surface instead of a plane, this equation won't work. This metric equation is ''only'' valid on a ''flat'' surface with ''Cartesian'' coordinates.
Which makes it pretty useless, since so much of physics revolves around curved spacetime and spherical coordinates.
What we need is a ''generalized'' metric equation, some way of measuring the interval of any two points regardless of what coordinate system we're using or whether our local geometry is flat or curved.
The ''metric tensor equation'' provides this generalization.
If <math>v</math> is any vector having components <math>v^\mu</math>, the length of <math>v</math> is given by the following equation:
:<math>
s^2 = g_{\mu\nu} v^{\mu} v^{\nu}\,
</math>
where <math>g_{\mu\nu}</math> is the ''metric tensor,'' and <math>\mu</math> and <math>\nu</math> range over the number of dimensions. Recall that Einstein summation notation means that this is actually a sum over indices <math>\mu</math> and <math>\nu</math>. If we assume that we're in the two-dimensional Euclidean plane, the metric tensor equation expands to:
:<math>
s^2 = g_{11} v^{1} v^{1} + g_{12} v^{1} v^{2} + g_{21} v^{2} v^{1} + g_{22} v^{2} v^{2}\,
</math>
The terms of the metric tensor, then, must be numerical coefficients in the metric equation. We already know what these equations need to be to make the metric equation work in the Euclidean plane with Cartesian coordinates:
:<math>
s^2 = (1) v^{1} v^{1} + (0) v^{1} v^{2} + (0) v^{2} v^{1} + (1) v^{2} v^{2}\,
</math>
Now we can write the metric tensor for the Euclidean plane in Cartesian coordinates in the form of a 2 × 2 matrix:
:<math>
g_{\mu\nu} = \begin{pmatrix}
1 & 0 \\
0 & 1
\end{pmatrix}
</math>
So in the case of the Euclidean plane with Cartesian coordinates, the metric tensor is the [[Kronecker delta]]:
:<math>
\delta^i_j = \begin{cases}
1, & \mbox{if }i = j \\
0, & \mbox{if }i \ne j
\end{cases}
</math>
Of course, the same concepts apply if we expand our interest from the plane to three-dimensional Euclidean ''space'' with Cartesian coordinates. We just have to let the indices of the Kronecker delta run from 1 to 3.
:<math>
g_{\mu\nu} = \begin{pmatrix}
1 & 0 & 0\\
0 & 1 & 0\\
0 & 0 & 1
\end{pmatrix}
</math>
Which gives us the following metric equation for the length of a vector <math>v</math> (omitting terms with zero coefficient):
:<math>
s^2 = (v^1)^2 + (v^2)^2 + (v^3)^2\,
</math>
Which precisely agrees with the Pythagorean theorem in three dimensions. So given a metric tensor <math>g_{\mu\nu}</math> for any space and coordinate basis, we can calculate the distance between any two points. The ''metric'' tensor, therefore, is what allows us to ''measure'' curved space. In a very real sense, the metric tensor describes the ''shape'' of both the underlying geometry and the chosen coordinate basis.
But relativity is concerned not with geometrically abstract ''space;'' we're interested in very real space''time,'' and that requires a slightly different kind of metric.
====The local Minkowski metric====
