Section 5.10 rewrite

5.10 Metric tensor: covariant scalar product and norm For a Cartesian space, Chapter 4 defined the norm as the length of a vector, the metric as the distance between two vectors, and the scalar product (inner product) as the projection of one vector on another. The official definitions of norm, metric and scalar product require non-negativity: |x| ≥ 0, d(x,y) ≥ 0, and x x ≥ 0. For non-Cartesian spaces, the logical extensions of these three concepts can result in all three quantities being negative. Nevertheless, we shall use the term "covariant scalar product" with notation A B as defined below, as well as the notation |A|2 ≡ A A where |A| will be called the length, magnitude or norm of A, even though these objects are not true scalar products or norms. In the curvilinear application of tensor analysis, where x-space is Cartesian, since the norm and scalar product are tensorial scalars, and since they are non-negative in Cartesian x-space, the problem of negative norms does not arise in either space. How do authors handle this problem? Some authors refer to A A as "the norm" of A (e.g., Messiah last line of p 878 discussing special relativity), which is our |A|2. For a general 4-vector A in special or general relativity, most authors just write A A (AμAμ in standard notation), they note that the quantity is invariant under transformations, but don't give it a name. Whereas we use the bold for this covariant dot product, most special relativity authors prefer to reserve this bold dot for a 3D spatial dot product, and then the 4D dot product is written with some "less bold dot" such as A.B or A•B. Typical usage then in standard notation would be p•p = pμpμ = p02 - pp (see for example Bjorken and Drell p 281). Without further ado, we define the "covariant scalar product" of two contravariant vectors (a new and different use of the word "covariant", but the same as appears in Section 7.15) as A B ≡ Aaa . // implied sum on a (5.10.1) The important fact about the dot product of two tensorial vectors is that it is a tensorial scalar, as we now show using (5.9.2) for the transformations of A and B and (2.1.6) that SR = 1 : A' B' ≡ A'aa' = (RA)a(ST)a = (RaiAi)(Sjaj) = (SjaRai)Aij = (SR)jiAij = δj,iAij = Aii = A B . (5.10.2) There are various equivalent ways to write the dot product of (5.10.1) using (5.8.4) that = V and conversely that V = g , and also the fact (5.4.3) that gab = gba : A B ≡ Aaa = Aa(abBb) = abAaBb = (baAa)Bb = bBb = aBa A B ≡ Aaa = (g)aa = gab b a = gab b a To summarize A B = Aaa = aBa = abAaBb = gab b a (5.10.3) In the special case that A = B, we use the shorthand norm notation (with caveat as noted above) and (5.10.2) to obtain, |A|2 ≡ A A = A' A' = |A'|2 . (5.10.4) Going back to Chapter 3 and the vectors e'n and en, a claim made at the start of Section 3.4 can now be verified: |e'n|2 = e'n e'n = en en = |en|2 => |e'n| = |en| . (5.10.5) Comment on Notation Consider again the definition (5.10.1) and the alternate form aBa shown in (5.10.3) A B ≡ Aaa = aBa . (5.10.3) The dot product involves the contravariant components of one vector and the covariant components of the other vector. In the dot product notation A B , we have indicated each vector by its contravariant name just as a convention. We could just as well have indicated one or both vectors by its covariant name, but the dot product indicated by whatever name would be the same: contravariant components of one vector and the covariant components of the other vector. Thus, A B ≡ Aii = iBi = B = A = . (5.10.6) We shall always use the first notation A B since it is the simplest. We then have the interesting fact that, if A and B are tensorial vectors for general x' = F(x), then A ≠ but A B = B for any B (5.10.7) If B = A, this says A ≠ but A A = A = |A|2 = Aii = gijAiAj . (5.10.8) We mention this notational issue to head off the following incorrect notion: A B = abAaBb = abab ≠ A B // wrong!!! A dot product application In applications in which (ds)2 is regarded as a scalar with respect to transformation F we have (ds')2 = dx' dx' = (ds)2 = dx dx (5.10.9) and ds = ds' is called "the invariant distance". Such applications include curvilinear coordinate transformations and relativity transformations. In special relativity, using the Bjorken and Drell notation noted above where g'μν = diag(1,-1,-,1,-1) and c = 1, one writes (Standard Notation), (dτ)2 = g'μν dx'μdx'ν = dx'μdx'μ = dx'• dx' = dx • dx = a Lorentz scalar = (dt)2 - dx dx , xμ = (t,x) (5.10.10) and dτ is called "the proper time", a particular case of the invariant distance ds. Notice that (dτ)2 < 0 for a spacelike 4-vector dxμ, meaning one that lies outside the future and past lightcones (|dx| > |dt| ). [We now restore to our covariant definition after temporarily using it above for a 3-space Cartesian dot product. ]