retired section 5_10

Retired Section 5.10 on dot products PhL 4.17.15 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 Chapter 7 (u) ) as sum a,b reorder a↔b sym A B ≡ abAaBb = abBbAa = baBaAb = abBaAb = B A (5.10.1) where we have used the fact shown in Section 5.4 that g is symmetric. This covariant scalar (or dot) product is more interesting and useful than the object AaBa because the covariant scalar product of two contravariant vectors is a tensorial scalar, as we now demonstrate (Picture A) : A' B' = 'abA'aB'b = 'ab(Raa'Aa') (Rbb'Bb') // (2.5.1) twice = 'ab Raa' Rbb' Aa' Bb' = [ (RT)a'a 'ab Rbb' ] Aa' Bb' = [RT ' R]a'b' Aa' Bb' = a'b' Aa' Bb' // (5.7.7) second line = ab Aa Bb = A B . (5.10.2) Recall from (5.8.4) that for any contravariant vector B, there is a partner covariant vector a = abBb. Using this partner one can restate the above covariant scalar product as A B = abAaBb = Aaa (5.10.3) or, taking instead b = baAa , A B = b Bb = a Ba . (5.10.4) And finally, if in A B = Aaa we write Aa = gabb , we get A B = Aaa = gabba = gabab (5.10.5) where the scalar product is now expressed in terms of the covariant partner vectors and along with the contravariant metric tensor g. To summarize, there are four different ways to write this covariant scalar product : A B = abAaBb = Aaa = aBa = gabab = B A . (5.10.6) 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 . (5.10.7) 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 (5.10.8) 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) (5.10.9) 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. ] Going back to Chapter 3 and the vectors e'n and en, a claim made there in (b) below (3.4.3) can now be verified: |e'n|2 = e'n e'n = en en = |en|2 => |e'n| = |en| . (5.10.10) Comment: The covariant dot product A B is a shorthand notation for any of these expressions abAaBb = Aaa = aBa = gabab It can be expressed in terms of all contravariant, all covariant, or mixed components as shown. The dot product A B is not "the dot product of two contravariant vectors", it is "the dot product of two tensorial vectors" which can be enumerated in the ways shown above. Although we have used the notation = V in (5.8.1) as a shorthand for i = ijVj to relate the two kinds of components, there really is only one vector V, there are not two vectors V and . For example, later in (6.6.9) we will show two expansions for this vector V, one of which involves the contravariant components Vn, and the other of which involves the covariant components n. V = V1 u1 + V2 u2 +... = ΣnVn un where Un V = Vn Un = gni ui V = 1 U1 + 2 U2 +... = Σnn Un where un V = n We never show any expansions for a vector because there is no vector to expand. V = Vn un V = Vn un (un)i = ik(un)k to describe how the components of a covariant vector are related to those of the corresponding contravariant vector, there is really only one vector V and it has these two kinds of components. = abAaBb = Aaa = aBa = gabab Need a comment about the meaning of this object