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a b, which gives Q ab Rab =
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1 1 Q ab Rab Q ab Rba = Q ab Rab Q ba Rab 2 2
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Now use the symmetry of Q ab to change the second term, giving the desired result: 1 1 1 Q ab Rab Q ba Rab = Q ab Rab Q ab Rab = Q ab (Rab Rab ) = 0 2 2 2 EXAMPLE 3-8 Show that if Q ab = Q ba is a symmetric tensor and Tab is arbitrary, then Tab Q ab = SOLUTION 3-8 Using the symmetry of Q, we have Tab Q ab = Tab 1 1 Q ab + Q ab = Tab Q ab + Q ba 2 2 1 ab Q (Tab + Tba ) 2
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Now multiply it by T , which gives 1 1 Q ab + Q ba = Tab Q ab + Tab Q ba 2 2
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Again, the indices a, b are repeated in both expressions. Therefore, they are dummy indices that can be changed. We swap them in the second term a b, which gives 1 1 1 Tab Q ab + Tab Q ba = Tab Q ab + Tba Q ab = Q ab (Tab + Tba ) 2 2 2
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The Levi-Cevita Tensor
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No book on relativity can really give you a good enough headache without mention of our friend, the Levi-Cevita tensor. This is +1 for an even permutation of 0123 (3.7) abcd = 1 for an odd permutation of 0123 0 otherwise We will see more of this in future chapters.
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1. If T = T then (a) Q ab Tab = 1 Q ab + Q ba Tab 2 (b) Q ab Tab = 1 Q ab Q ba Tab 2 (c) Q ab Tab = 1 Q ab Q ba Tab 2 (d) Q ab Tab = 1 (Q ab Q ba ) Tab 2 Let x a ( ) be a parameterized curve. The components of the tangent vector to this curve are best described by (a) d /dx (b) f = x ( ) (c) dx a /d (d) x = f ( ) A tensor with components Ta b c has an expansion in terms of basis vectors and one forms, given by (a) T = Ta b c a b c (b) T = Ta b c a eb c (c) T = Ta b c ea b ec The symmetric part of Va Wb is best written as (a) V(a Wb) = 1 (Va Wb Wb Va ) 2 (b) V(a Wb) = 1 V a Wb + W b Va 2 (c) V(a Wb) = 1 (Va Wb + Wb Va ) 2 The Kronecker delta acts as a (a) b T bc d = T ac d a (b) b T bc d = Tacd a (c) b T bc d = Ta c d a (d) b T bc d = T ac d
ab ba
CHAPTER
Tensor Calculus
In this chapter we turn to the problem of nding the derivative of a tensor. In a curved space or spacetime, this is a bit of a thorny issue. As we will see, properly nding the derivative of a tensor, which should give us back a new tensor, is going to require some additional mathematical formalism. We will show how this works and then describe the metric tensor, which plays a central role in the study of gravity. Next we will introduce some quantities that are important in Einstein s equation.
Testing Tensor Character
As we will see below, it is sometimes necessary to determine whether a given object is a tensor or not. The most straightforward way to determine whether an object is a given type of tensor is to check how it transforms. There are, however, a few useful tips that can serve as a guide as to whether or not a given quantity is a tensor. The rst test relies on the inner product. If the inner product a V a =
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