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Copyright 2006 by The McGraw-Hill Companies, Inc. Click here for terms of use.
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The Energy-Momentum Tensor
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crossing the interface de ned by constant x b . In this case we re talking about the momentum four vector, so if a = t then we are talking about the ow of energy across a surface. Let s describe each type of component we can have in turn. These are T tt , T it , T ti , and T ij .
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The T tt component represents energy density. To see why, consider the momentum four vector such that p = (E, p). Using the de nition we gave above, we see that in this case T tt is the p 0 component of the momentum four vector, or simply the energy, crossing a surface of constant time. This is energy density. In relativity, energy and mass are equivalent, and so we should really think of this as the mass-energy density. In most applications energy density is denoted by u; however, we don t want to confuse that with the four velocity and so we denote the density of mass-energy by . Therefore, for the stress-energy tensor, we can write = T tt
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Momentum density is momentum per unit volume. If we call momentum density , then the momentum density in the idirection is i = T it This is the ow of momentum crossing a surface of constant time. Now consider T ti . This term (which is actually equal to T it since the energy-momentum tensor is symmetric) represents the energy ow across the surface x i .
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The nal piece of the stress-energy tensor is given by the purely spatial components. These represent the ux of force per unit area which is stress. We have T ij
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time time component = energy density
time space components = energy flux
= T ab
space time components = momentum density spatial components represent stress
Fig. 7-1. A schematic representation of the stress-energy tensor. T 00 is the energy
density. Terms of the form T 0j (where j is a spatial index) are energy ux. Terms with T j0 are momentum density, while purely spatial components T ij are stress.
This term is the ith component of force per unit area (which is stress) across a surface with normal direction given by the basis vector e j . Analogously, T ji is the jth component of force per unit area across a surface with normal given by the basis vector ei . Returning to the view of a tensor that maps vectors and one forms to the real numbers, we obtain these components of the stress-energy tensor by passing as argument the basis vectors; i.e., Tij = T ei , e j The organization of the components of the stress-energy tensor into a matrix is shown schematically in Fig. 7-1. We will consider two types of stress-energy tensor seen frequently in relativity: perfect uids and dust.
Conservation Equations
Conservation equations can be derived from the stress-energy tensor using b T ab = 0 (7.1)
The Energy-Momentum Tensor
This equation means that energy and momentum are conserved. In a local frame, this reduces to T ab =0 xb (7.2)
In the local frame, when the conservation law (7.2) is applied to the time coordinate we obtain the familiar relation: T 00 T 0i T 00 T 0i + + + =0 = = i i t x t x t which is the conservation of energy.
Dust
Later we will describe a perfect uid which is characterized by pressure and density. If we start with a perfect uid but let the pressure go to zero, we have dust. This is the simplest possible energy-momentum tensor that we can have. It might seem that dust is too simple to be of interest. However, consider that the dust particles carry energy and momentum. The energy and momentum of the moving dust particles give rise to a gravitational eld. In this case, there are only two quantities that can be used to describe the matter eld in the problem the energy density and how fast (and in what direction) the dust is moving. The simplest way to obtain the rst quantity, the energy density, is to jump over to the co-moving frame. If you re in the comoving frame, then you re moving along with the dust particles. In that case there is a number of dust particles per unit volume n, and each dust particle has energy m. So the energy density is given by = mn. The second item of interest is none other than the velocity four vector u. This of course will give us the momentum carried by the dust. Generally speaking, to get the stress-energy tensor for dust, we put this together with the energy density. So for dust, the stress-energy tensor is given by T ab = u a u b (7.3)
For a co-moving observer, the four velocity reduces to u = (1, 0, 0, 0). In this case, the stress-energy tensor takes on the remarkably simple form
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