The Metric Tensor

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

(Eq地 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地 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地 3)

    If we multiply Equation 2 by itself, we get

(Eq地 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地 5)

Finally we remove the imaginary coefficients from the temporal terms and multiply the M44 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地 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地 7)

In that equation the 4x4 matrix gik 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地 8)

For example, in plane polar coordinates we have contravariantly

(Eq地 9)

and covariantly we have

(Eq地 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地 11)

We calculate the variation of Equation 7 to get

(Eq地 12)

which, if we take xs as the coordinate measured along the path s, gives us

(Eq地 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地 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地 15)

    Adding the acceleration term to both sides of that equation and multiplying the result by the contravariant metric tensor gms gives us

(Eq地 16)

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

(Eq地 17)

and also used the definition of the Christoffel symbol,

(Eq地 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/fL, in which the Lorentz factor takes the form

(Eq地 19)

Thus we have Equation 16 in the form

(Eq地 20)

in which

(Eq地 21)

In all four dimensions

(Eq地 22)

so we have

(Eq地 23)

In components we have Equation 20 as

(Eq地s 24)

Using the fourth of those equations to replace fLτfL in the first three and then subtracting the result from both sides gives us

(Eq地 25)

which gives us the three spatial components of acceleration (we ignore the temporal component). Taking βμ=fLvμ/c (vμ=(vi,c)), we can cancel (fL)2 and 1/c2 out of the equation to obtain the measurable acceleration as

(Eq地 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地 8)

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

(Eq地 27)

Substituting from Equation 8 gives us

(Eq地 28)

We require that dxm=dxk, so we have

(Eq地 29)

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

(Eq地 30)

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

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

(Eq地 31)

In that equation we have

(Eq地 32)

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

(Eq地 33)

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

(Eq地s 34)

We thus devise the contravariant Schwarzschild metric tensor as

(Eq地 35)

    As an example of how Equation 18 works for us, let痴 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 g11 and g44 elements. The elements g22 and g33 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地s 36)

The relevant Christoffel symbols in this case thus consist of

(Eq地s 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地 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痴 law of gravity. But, we must note, we originally started with Newton痴 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地 39)

On first impression it doesn稚 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地 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地 41)

    Next Einstein pointed to the transformation of the differential element of 4-volume dV4=dxdydzcdt. By Jacobi痴 theorem we get

(Eq地 42)

If we multiply Equations 41 and 42 together, we get

(Eq地 43)

Thus we obtain another relativistic invariant.

    In Minkowski space g=-1, so if dV4' represents the element of Minkowskian 4-volume, we have

(Eq地 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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