Special Relativity II—standard notation

©Fernando Caracena 2013

At first, Einstein did not like his teacher's representation of four-vectors, but later he realized that the notation allowed him to generalize Special Relativity into what he later called General Relativity, which was a theory that incorporated the effects of gravity on the geometry of space and time. General Relativity is then a field theory of gravity.

The metric tensor

The vector notation that the formulation of Relativity uses involves both subscripts and superscripts, which are not to be confused with exponents. Also, the concept of vectors is generalized to quantities that are indexed by several vector indices, which are called tensors. The central tensor involved in Special Relativity in Cartesian coordinates is called the metric tensor (gμν), which has every element equal to zero, except the diagonal elements, which are either +1 or -1 as follows:

,(1a)

where the first index (μ) represents the row number starting from the top(μ=0) and ending on the bottom row (μ= 3), and the second (ν), the column number beginning on the left (ν = 0) and ending on the right       (ν =3). Those using the metric tensor sometimes use what is the negative of the metric tensor (1a) so that the negative sign is confined to the upper left component (g00=-1) and the remaining diagonal components given the value +1. The results are not changed by the choice of the sign of the diagonal elements, so long as one sticks to the same notation; but in reading someone's work in Relativity, on should be aware of what convention that person is using.

The zeroth index value (μ=0) refers to the time component of a four-vetor or four-tensor, whereas the other three index values (μ=1, 2, 3) refer to  the spatial components.

Mathematicians have invented a notation for the equality of indexes in a what is called the Kronecker delta, which is defined as follows in four dimensions:

δμν = 0 if μ ≠ ν

and

δμν = 1 if μ = ν          .                                                                                                (1b)

We can define the metric tensor in terms of the Kronecker as follows,

,                                                                                          (1c)

where i is that part of the range of μ that is not zero, and j, the same for ν.

The inverse of the metric tensor is defined as the same symbol as the metric tensor, but with superscripts,

gμν = inverse(gμν)

gμν = gμν        ,                                                                                                               (1d)

which in this case is the same as the metric tensor itself. Note that (1d) is not a general property of the metric tensor, but is true only in Special Relativity written in Cartesian coordinates.

Scalar product

Now we are ready to flesh out the notation of the scalar product of two four-vectors, which is the product of the corresponding components of a covariant (lower indexed) vector with a contravariant (upper indexed) vector respectively.

,                                                                                      (2a)

In other words, the scalar product always involves a covariant and a contravariant vector summed over the product of their respective components.

Rather than carrying an explicit summation symbol, such as in (2a), Einstein proposed using a summation convention, in which repeated combinations of covariant and contravariant indices automatically implied a sum over them of all four components (0,1,2,3).  Applied as in (2), the summation convention has a cleaner look,

P•P = Pμ Pμ   (summation over repeated indices implied).      (2b)

If we take the contravariant vector as having all positive components,

Pμ = (P0, P1, P2, P3) ,                                                                (2c)

then the covariant vector has the same components, but the spatial parts have reversed signs,

Pμ = (P0 , -P1, -P2, -P3).                                                             (2d)

In this case the sum over the four indices in (2b) gives the scalar product of four-vectors that was used in part I,

Pμ Pμ = (P0)2 -(P1)2 -(P2)2 -(P3)2.                                                (2e)

Note that we have used parentheses around each vector component when raising it to some power [as in (2e)] to distinguish that power from the superscript of that vector component.

It is left as an exercise for the reader to show that the scaler product of a 4-mementum vector with itself is the following invariant:

Pμ Pμ= (m0c)2 .                                                                         (2f)

Raising and lowering indices

Raising an index in a four-vector (P) is equivalent to transforming a covariant vector (Pv) into a contravaiant vector (Pμ). This transformation is accomplished by a matrix operation, which is equivalent to the scalar product taken over the second index of the metric tensor ( gμv) and the contravariant index of    (Pv)

Pμ = gμv Pv .                                                                                                                  (3a)

It is left as an exercise to the reader that (3a) results in (2d) given that the metric tensor is given by (1a).

Because the lowering of an index is also done by contracting the vector with the metric tensor, the following shows that the metric tensor is also its own inverse:

Pμ = gμα gαv Pv  ,                                                                                                             (3b)

writing the identity,

Pμ = δμν Pv   ,                                                                                                                  (3c)

identifies what the product of the metric tensor is

gμα gαv = δμν     .                                                                                                              (3d)

The Lorentz Transformation revisited

,                                                                  (4)

where

γ =1/√(1-β2)                                                                                                                (4a)

and

β = v/c  .                                                                                                                      (4b)

In matrix notation, we can write the position 4-vector (Pμ) and the Lorentz transformed position vector (Pμ') as follows:

,                                                                                                     (5a)

and

.                                                                                                     (5b)

The Lorentz transformation written as a matrix equation appears as follows:

R' = ΛR  ,                                                                                                                  (5c)

where the product on the RHS of (5c)  is a matrix product of the matrix (Λ) defined in (4)  with the position vector (R) defined in (5a) . It is left as an exercise for the reader to carry out the matrix multiplication to show that it gives the same results as in Grokking Special Relativity (GSR) in the set of equations (GSR.L.1a)- (GSR.L.4a).

All tensors in Special Relativity, transform on each index the same way that vectors do. For example, let us transform the metric tensor into the primed coordinate system, as follows:

g' μν μ α Λν β gα β              .                                                                                           (6a)

By reversing the rows and columns on the second Lorentz matrix on the RHS of (6a), and undoing the same with the transpose of the resulting matrix, one can write the transformed metric tensor as,

g'μν =Λμ α gα β (Λ βν)T  ,

which in matrix notation appears as follows:

g' = Λ • g ΛT      .                                                                                                     (6c)

The Lorentz transformation matrix happens to be symmetric,

ΛT   = Λ ,                                                                                                                      (6d)

so that we can write the simpler expression,

g' = Λ • g Λ      .                                                                                                       (6e)

It is left as an exercise for the interested reader to show that (6e) is true, and that g' has the same form as g . In general, a tensor having any number of superscripts in combination with subscripts (Tα β γ...δ) transforms on each superscript the same way that a vector transforms under a Lorentz Transformation

T' μνσ...τμαΛν β Λσγ … Λ τδTα β γ...δ      .                                                                   (7)

Conclusion

This discussion centers on the formalism of Relativity, which allows the reader to connect with most of its technical discussions found elsewhere. Although the formalism appears bulky, it nevertheless lends itself to being generalized to include the effects of gravity, which modify the metric through concentrations of mass-energy. In The General Theory of Relativity, these effects through curvature of space-time change the metric from being a set of constant components to being functions of the distributions of mass-energy. Technically, General Relativity is an area of application of the mathematical theory of differential geometry in curvilinear coordinates, which unfortunately cannot be globally reduced to Cartesian coordinates. If it could, it would not be necessary to carry such a bulky apparatus of of indexed quantities. Studying General Relativity is like climbing Mount Everest. It takes a lot of preparation, acclimatization, and the learning how to use various pieces of hardware such as pitons, ropes and ice axes.

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