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# 9.E: Flux (Exercises)

• • Contributed by Benjamin Crowell
• Professor (Physics) at Fullerton College

## Q1

Rewrite the stress-energy tensor of a perfect ﬂuid in SI units. For air at sea level, compare the sizes of its components.

## Q2

Prove by direct computation that if a rank-$$2$$ tensor is symmetric when expressed in one Minkowski frame, the symmetry is preserved under a boost.

## Q3

Consider the following change of coordinates:

$t' = -t$

$x' = x$

$y' = y$

$z' = z$

This is called a time reversal. As in Example 9.2.3, ﬁnd the eﬀect on the stress-energy tensor.

## Q4

Show that in Minkowski coordinates in ﬂat spacetime, all Christoﬀel symbols vanish.

## Q5

Show that if the diﬀerential equation for geodesics is satisﬁed for one aﬃne parameter $$λ$$, then it is also satisﬁed for any other aﬃne parameter $$λ' = aλ+b$$, where $$a$$ and $$b$$ are constants.

## Q6

This problem investigates a notational conﬂict in the description of the metric tensor using index notation. Suppose that we have two diﬀerent metrics, $$g_{µν}$$ and $$g'_{µν}$$. The diﬀerence of two rank-$$2$$ tensors is also a rank-$$2$$ tensor, so we would like the quantity $$\partial g_{\mu \nu } = g'_{\mu \nu } - g_{\mu \nu }$$ to be a well-behaved tensor both in its transformation properties and in its behavior when we manipulate its indices. Now we also have $$g_{µν}$$ and $$g'_{µν}$$, which are deﬁned as the matrix inverses of their lower-index counterparts; this is a special property of the metric, not of rank-$$2$$ tensors in general. We can then deﬁne $$\partial g^{\mu \nu } = g'^{\mu \nu } - g^{\mu \nu }$$.

1. Use a simple example to show that $$\partial g_{\mu \nu }$$ and $$\partial g^{\mu \nu }$$ cannot be computed from one another in the usual way by raising and lowering indices.
2. Find the general relationship between $$\partial g_{\mu \nu }$$ and $$\partial g^{\mu \nu }$$.

## Q7

In section 9.5, we analyzed the Bell spaceship paradox using the expansion scalar and the Herglotz-Noether theorem. Suppose that we carry out a similar analysis, but with the congruence deﬁned by $$x^2 - t^2 = a^{-2}$$. The motivation for considering this congruence is that its world-lines have constant proper acceleration $$a$$, and each such world-line has a constant value of the coordinate $$X$$ in the system of accelerated coordinates (Rindler coordinates) described in section 7.1. Show that the expansion tensor vanishes. The interpretation is that it is possible to apply a carefully planned set of external forces to a straight rod so that it accelerates along its own length without any stress, i.e., while remaining Born-rigid.