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Vectors, One Forms, Metric
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the indices i, j will range only over (1, 2, 3). Typing a lot of Greek symbols is a bit of extra work, so we will stick to using Latin indices all the time. When possible, we will use early letters (a, b, c, . . .) to range over all possible values (0, 1, 2, 3) and use letters from the middle of the alphabet such as i, j to range only over the spatial components (1, 2, 3).
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The Einstein Summation Convention
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The Einstein summation convention is a way to write sums in a shorthand format. When the same index appears twice in an expression, once raised and once lowered, a sum is implied. As a speci c example,
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Ai B i Ai B i = A 1 B 1 + A 2 B 2 + A 3 B 3
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Another example is that S a Tab is shorthand for the expression
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S a Tab
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An index that is summed over is called a dummy index, and can be replaced by another label if it is convenient. For example, S a Tab = S c Tcb The index b in the previous expressions is not involved in the sum operations. Such an index is known as a free index. A free index will typically appear on both sides of an expression. For example, consider the following equation: Aa =
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a bA b
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In this expression, b is once again a dummy index. The sum implied here means that Aa =
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a bA b
The other index found in this expression, a , is a free index. If we elect to change the free index, it must be changed on both sides of the equation. Therefore it
Vectors, One Forms, Metric
would be valid to make the change a b , provided that we make this change on both sides; i.e., Ab =
b bA b
Tangent Vectors, One Forms, and the Coordinate Basis
We will often label basis vectors with the notation ea . Using the Einstein summation convention, a vector V can be written in terms of some basis as V = V a ea In this context the notation ea makes sense, because we can use it in the summation convention (this would not be possible with the cumbersome (i, j, k) for example). In a given coordinate system, the basis vectors ea are tangent to the coordinate lines. (See Fig. 2-3 and Fig. 2-4) This is the reason why we can write basis vectors as partial derivatives in a particular coordinate direction (for an explanation, see Carroll, 2004). In other words, we take ea = a = xa
This type of basis is called a coordinate basis. This allows us to think of a vector as an operator, one that maps a function into a new function that is related to its derivative. In particular, V f = (V a ea ) = V a a f A vector V can be represented with covariant components Va . This type of vector is called a one form. Basis one forms have raised indices and are often denoted by a . So we can write V = Va a We have used a tilde to note that this is a one form and not an ordinary vector (but it is the same object, in a different representation). Later, we will see how to move between the two representations by raising and lowering indices with the
Vectors, One Forms, Metric
Fig. 2-3. A tangent vector to a curve.
metric. The basis one forms form a dual vector space to ordinary vectors, in the sense that the one forms constitute a vector space in their own right and the basis one forms map basis vectors to a number, the Kronecker delta function; i.e.,
a a (eb ) = b
(2.1)
where
a b =
a=b otherwise
In a coordinate representation, the basis one forms are given by a = dx a (2.2)
With this representation, it is easy to see why (2.1) holds. An arbitrary one form a maps a vector V a to a number via the scalar product V = a V a We can think of this either way: we can visualize vectors as maps that take one forms to the real numbers via the scalar product. More generally, we can de ne a ( p, q) tensor as a function that takes p one forms and q vectors as input and
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