The Metric Tensor

In Special Relativity we sum up the Lorentz Transformation in the Minkowski metric equation,

(Eq’n 1)

That equation gives us the four-dimensional equivalent of the Pythagorean theorem, describing the distance between pairs of events rather than between pairs of points. Where spacetime conforms to that equation we say that it manifests the Minkowski metric.

A proper understanding of the metric tensor comes from the fact, expressed in Equation 1, that the dot product of two four-vectors remains invariant under a Lorentz Transformation. Using a transformation matrix, we writ e the basic Lorentz Transformation like this:

(Eq’n 2)

In that equation β=v/c, with the relative velocity assumed to run parallel to the x-axis of our coordinate frame. Also I have put the imaginary coefficient (the square root of minus one) on the time coordinate so that the Minkowski metric equation will have a minus sign in the right place when we multiply the displacement four-vector by itself. The imaginary coefficient also appears in the transformation matrix to convert imaginary numbers into real numbers and vice versa as needed. We also have the Lorentz factor,

(Eq’n 3)

If we multiply Equation 2 by itself, we get

(Eq’n 4)

We can simplify the right side of that equation by reversing the order of the first two factors. In carrying out that reversal we must transpose both factors; that is, we must rewrite the column vector as a row vector and flip the matrix about its upper-left to lower-right diagonal. Then we have

(Eq’n 5)

Finally we remove the imaginary coefficients from the temporal terms and
multiply the M_{44} component of the matrix by minus one to put the
equation into the conventional form. The matrix thus becomes the metric tensor
of the Minkowski spacetime.

We assume that the metric tensor tells us something profound about the mutual geometry of space and time. The transformation matrix tells us how to convert spatio-temporal measurements made in one inertial frame of reference into the equivalent measurements made in another frame. In that conversion we see the bizarre effects that we associate with Special Relativity. The fact that multiplying the Lorentz Transformation matrix by its own transpose yields a metric tensor identical to the identity matrix reveals that Minkowski spacetime, in spite of the bizarre effects (such as time dilation) that we see in the Lorentz Transformation, is analogous to a flat plane.

If we express the measurements of distance and duration between two events as a four-vector, (dx,dy,dz,cdt), then we can write Equation 1 as

(Eq’n 6)

Thus we get the four-dimensional dot product of the differential-interval four-vector with itself. The matrix represents what we call the metric tensor. In this case its significant action merely puts the minus sign in the sum, which seems rather trivial. In General Relativity the matrix becomes more active.

We can also write Equation 1 in tensor notation as

(Eq’n 7)

In that equation the 4x4 matrix g_{ik} represents the metric tensor.
The contravariant variables (superscripted) represent what we measure directly
and the covariant variables (subscripted) represent what we obtain by inference
through the metric tensor,

(Eq’n 8)

For example, in plane polar coordinates we have contravariantly

(Eq’n 9)

and covariantly we have

(Eq’n 10)

The metric tensor finds little use in Special Relativity, but in General Relativity it plays a central role in the equations of motion. That latter theory emphasizes the proposition that space itself moves and that matter merely rides along. Thus, the principle of least action becomes the principle of minimum displacement,

(Eq’n 11)

We calculate the variation of Equation 7 to get

(Eq’n 12)

which, if we take x^{s} as the coordinate measured along the path s,
gives us

(Eq’n 13)

In going from the first line in that equation to the second I replaced the last term with its equivalent through the product rule of differentiation and I factored out the ds. In the last two terms on the second line I replaced the k-index with the s-index for convenience: because we sum both indices over all of their possible values, the replacement makes no difference in the result. The middle term on the second line gives us a perfect differential, as in the standard derivation of the Euler-Lagrange equations, so we can integrate it immediately and evaluate it at the endpoints of the domain of integration. The variation goes to zero at those endpoints, so the whole term vanishes and we get

(Eq’n 14)

Of course that works because the operations of variation and integration commute with each other. As usual, the variation in the coordinates is purely arbitrary, so the factor enclosed in square brackets must equal a perfect zero. That fact gives us

(Eq’n 15)

Adding the acceleration term to both sides of that
equation and multiplying the result by the contravariant metric tensor g^{ms}
gives us

(Eq’n 16)

In producing that equation I have made use of the fact that

(Eq’n 17)

and also used the definition of the Christoffel symbol,

(Eq’n 18)

So Equation 16 stands as the equation of motion in warped spacetime.

To convert Equation 16 into a description of the actual
coordinate acceleration we make the substitution ds=cdt/f_{L}, in which
the Lorentz factor takes the form

(Eq’n 19)

Thus we have Equation 16 in the form

(Eq’n 20)

in which

(Eq’n 21)

In all four dimensions

(Eq’n 22)

so we have

(Eq’n 23)

In components we have Equation 20 as

(Eq’ns 24)

Using the fourth of those equations to replace f_{L}∂_{τ}f_{L}
in the first three and then subtracting the result from both sides gives us

(Eq’n 25)

which gives us the three spatial components of acceleration (we ignore the
temporal component). Taking β^{μ}=f_{L}v^{μ}/c
(v^{μ}=(v^{i},c)),
we can cancel (f_{L})^{2} and 1/c^{2} out of the
equation to obtain the measurable acceleration as

(Eq’n 26)

One component that we need in order to calculate a Christoffel symbol is a contravariant version of the metric tensor. To obtain a contravariant metric tensor we must raise both of the indices of the covariant metric tensor, using something like the inverse of the procedure represented in Equation 8. For an arbitrary differential distance we have

(Eq’n 8)

We want to reverse that transformation, so we define the contravariant metric tensor by writing

(Eq’n 27)

Substituting from Equation 8 gives us

(Eq’n 28)

We require that dx^{m}=dx^{k}, so we have

(Eq’n 29)

in which the delta represents the Kronecker delta. Since the metric tensor is a square matrix, we can use Cramer’s rule to find the contravariant metric tensor as

(Eq’n 30)

The cofactor is itself a determinant, one that we can calculate by removing
the n-th row and the k-th column from g_{nk} and calculating the
determinant of the remaining matrix.

Let’s consider, for example, the metric tensor encoding the classic Schwarzschild solution of Einstein’s equation. The covariant metric tensor comes to us as

(Eq’n 31)

In that equation we have

(Eq’n 32)

in which M represents the mass of a spherically symmetric gravitating body. We calculate the determinant as

(Eq’n 33)

Because g_{nk} is a purely diagonal matrix, we only need to calculate
four of the cofactors; the other twelve all zero out. So we have

(Eq’ns 34)

We thus devise the contravariant Schwarzschild metric tensor as

(Eq’n 35)

As an example of how Equation 18 works for us, let’s
calculate the Christoffel symbols of the Schwarzschild metric. For simplicity we
will assume that motion occurs only in the radial direction, so we need only to
consider the derivatives of the g_{11} and g_{44} elements. The
elements g_{22} and g_{33} will have non-zero derivatives
certainly, but because dθ=0
and dϕ=0
they will simply drop out of our calculation. So we have only two derivatives to
use in Equation 18,

(Eq’ns 36)

The relevant Christoffel symbols in this case thus consist of

(Eq’ns 37)

All of the other Christoffel symbols equal zero.

Inserting those values into Equation 26 gives us the coordinate acceleration at r (distinct from the radial acceleration that a local observer would measure with local clocks and rulers). We thus have

(Eq’n 38)

For objects and observers moving very much slower than the speed of light, the gamma factor (the Schwarzschild analogue of the Lorentz factor) and the factor enclosed in parentheses come very close to equaling one and Equation 38 thus turns into a clear expression of Newton’s law of gravity. But, we must note, we originally started with Newton’s law to derive the Schwarzschild metric, so we have actually come in a logical circle. The fact that the circle closed, coming back to its original statement, tells us that the logic and the theory to which we applied it are sound. So now we see how warping a metric tensor produces the effect of a force.

Several functions of the metric tensor exist and give us information about the nature of spacetime.

One of those functions is the determinant,

(Eq’n 39)

On first impression it doesn’t look like it would do anything of value for us. But Einstein pointed out that something interesting happens when we apply a transformation of coordinates to the determinant;

(Eq’n 40)

Because the determinant of the metric tensor always comes out a negative number and we want to use real numbers in our descriptions of spacetime, we write

(Eq’n 41)

Next Einstein pointed to the transformation of the
differential element of 4-volume dV_{4}=dxdydzcdt. By Jacobi’s theorem
we get

(Eq’n 42)

If we multiply Equations 41 and 42 together, we get

(Eq’n 43)

Thus we obtain another relativistic invariant.

In Minkowski space g’=-1, so if dV_{4}' represents
the element of Minkowskian 4-volume, we have

(Eq’n 44)

The invariant equals the magnitude of the minuscule element of 4-volume at the given locus, based on distances and durations measured by a local observer with clocks and rigid rods. Because we use differentials, we define our differential coordinates as measurements in the manner of Special Relativity, using the assumption that on a small enough scale the spacetime is indistinguishable from the Minkowski spacetime.

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