The Energy-Momentum Tensor in Visual Studio .NET

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The Energy-Momentum Tensor
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0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0
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(7.4)
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Now consider the case of a stationary observer seeing the dust particles go by with four velocity u. In that case, we have u = ( , u x , u y , u z ) , where 1 the u i are the ordinary components of three velocity and = 1 v 2 . Looking at (7.3), we see that in this case the stress-energy tensor is 1 ux = 2 y u uz ux (u x )2 u yux uzux uy ux u y (u y )2 uzu y uz ux uz y z u u (u z )2
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(7.5)
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EXAMPLE 7-1 Show that the conservation equations for the energy-momentum tensor in the case of dust lead to the equation of continuity of a uid. SOLUTION 7-1 The conservation equation is given by T ab =0 xb Setting a = t, we obtain T tx T ty T tz T tt T tb + + + =0 = xb t x y z Using (7.5), this becomes T tt T tx T ty T tz ( u x ) ( u y ) ( u z ) + + + = + + + t x y z t x y z = + ( u) t
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and so we have
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The Energy-Momentum Tensor
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+ ( u) = 0 t where u is the ordinary three-dimensional velocity. This is the equation of continuity.
Perfect Fluids
A perfect uid is a uid that has no heat conduction or viscosity. As such the uid is characterized by its mass density and the pressure P. The stress-energy tensor that describes a perfect uid in the local frame is 0 = 0 0 0 P 0 0 0 0 P 0 0 0 0 P
T ab
(7.6)
To nd the form of the stress-energy tensor in a general frame, we rst consider the at space of special relativity and boost to a frame of an observer with four velocity u. The stress-energy transforms to a general frame by Ta b =
a c b dT cd
(7.7)
However, we note that we can construct the most general form of the stressenergy tensor from the four velocity u, the metric tensor ab along with and p. Furthermore, the tensor is symmetric. This tells us that the general form of the stress energy tensor is T ab = Au a u b + B ab (7.8)
where A and B are scalars. For this example, we assume that the metric is ab = diag (1, 1, 1, 1). Looking at (7.6), we notice that the only spatial components are T ii = P. Another way to write this is T ij = ij P (7.9)
The Energy-Momentum Tensor
In the rest frame, we have u 0 = 1 and all other components vanish. Therefore, (7.8) takes the form T ij = B ij Comparison with (7.9) leads us to take B = P. Now we consider the time component. In the local frame it is given by T 00 = and so T 00 = = Au 0 u 0 + B 00 = Au 0 u 0 P = A P Therefore, we conclude that A = P + and write the general form of the stress-energy tensor for a perfect uid in Minkowski spacetime as T ab = ( + P) u a u b P ab For any metric gab , this immediately generalizes to T ab = ( + P) u a u b Pgab (7.11) (7.10)
Note that the form of the stress-energy tensor in general will change if we take ab = diag ( 1, 1, 1, 1). In that case the equations become T ab = ( + P) u a u b + P ab T ab = ( + P) u a u b + Pgab EXAMPLE 7-2 Consider the Robertson-Walker metric used in Example 5-3: ds 2 = dt 2 + a 2 (t) dr 2 + a 2 (t)r 2 d 2 + a 2 (t)r 2 sin2 d 2 1 kr 2 (7.12)
Suppose we take the Einstein equation with nonzero cosmological constant. Find the Friedmann equations in this case. SOLUTION 7-2 In the last chapter we found that in the local frame, the components of the Einstein tensor for this metric were given by 3 k + a2 a2 1 a G r r = G = G = 2 2 k + a 2 a a G t t =
(7.13)
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