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I've been working through Wald's book "General Relativity", and I'm fairly stuck on question 4. a) of chapter 6:

"Let $(M,g_{ab})$ be a stationary space-time with time-like Killing field $\xi^a$. Let $V^2 = -\xi^a \xi_a$. Show that the acceleration $a^b = u^a \nabla_a u^b$ of a stationary observer is given by $a^b = \nabla^b \log (V)$."

Firstly I used that a stationary observer is one whose 4-velocity is the time-like Killing vector $\xi^a$, and I introduced coordinates $t,x,y,z$ so that $\xi^a = (\partial / \partial t)^a$ and the remaining three are arbitrarily defined. Then $\xi^a$ is the vector $(1,0,0,0)$ so \begin{align} a^b &= \xi^a \nabla_b \xi^a \\ &= \xi^a \partial_a \xi^b + \xi^a \Gamma^b \text{}_{ac} \xi^c \\ &= \xi^0 \partial_0 \xi^b + \xi^0 \Gamma^b \text{}_{00} \xi^0 \\ &= 0 + \Gamma^b \text{}_{00} \\ &= (1/2) g^{ab} ( 2 \partial_t g_{0a} - \partial_a g_{00}) \\ &= -(1/2) \nabla^b g_{00}, \end{align} where to reach the last line the independence of $g_{ab}$'s coordinates from the Killing coordinate $t$ and the property $\nabla_a f = \partial_a f$ for any scalar $f$ were both used. Furthermore \begin{align} V^2 = - \xi^a \xi_a = - g_{ab} \xi^a \xi^b = -g_{00}, \end{align} so in fact $a^b = (1/2) \nabla^b (V^2) = V \nabla^b V$. However, $\nabla^b \log (V) = (1/V) \nabla^b V$ by the chain rule, so it appears necessary and sufficient, for $a^b$ to equal $\nabla^b \log(V)$, that $V = \pm 1$; indeed it could only be $+1$ since otherwise $\log(V)$ is undefined. But this would imply the highly unphysical situation that $a^b = 0$.

Can anybody help me resolve my quandary? Many thanks in advance!

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1 Answer

up vote 1 down vote accepted
  • We require $u^a u_a = -1$ for 4-velocities, but $\xi^a \xi_a = -V^2$ is not generally like this.
  • $g_{00}$ is not a scalar, it's a component of a tensor.

Let's not mess about with specific frames.

We have: $V^2 = \xi^a\xi_a$, $a^b = u^a\nabla_a u^b$, $\nabla_{(a}\xi_{b)}=0$, $u^a = f \xi^a$.

Firstly, $u^a u_a = f^2 \xi^a \xi_a = -V^2 f^2$ so $f=\pm1/V$.

Let us write $$a_b = 1/V\,\xi^a\nabla_a(\xi_b/V) = \frac{1}{V^2}\xi^a\nabla_a\xi_b-\frac{1}{V}\xi^a\xi_b\nabla_a \frac{1}{V}$$ But then using the antisymmetry property $$\xi^a\nabla_a(\xi_b) = -\xi^a\nabla_b(\xi_a) = -\frac{1}{2}\nabla_b(\xi^a\xi_a) = \frac{1}{2}\nabla_b V^2$$ and again $$\xi^a\xi_b\nabla_a V^2 = -\xi^a\xi_b\nabla_a (\xi_c \xi^c) = -2\xi_b\xi^a\xi^c\nabla_a \xi_c = 0$$

Hence

$$a_b = \frac{1}{2V^2}\nabla_b V^2 = \nabla_b \log V$$

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Great, that's exactly the pointer I needed. I didn't think a stationary observer need only have 4-velocity that's a scalar multiple of the Killing field. Although I must say I disagree with your statement that $g_{00}$ is not a scalar: it may depend on the choice of coordinates, but it nonetheless assigns a number to each point, making it one of a family of scalar functions on this region of $M$. My calculation now works with the alteration $u^a = \pm \xi^a / V$. Thanks again! –  Stromael Apr 22 '13 at 22:29
    
The definition of a scalar unequivocally requires it to be invariant under coordinate transformations. Put another way, it exists in a coordinate free world too. –  Sharkos Apr 22 '13 at 22:51
    
For example, $\partial_a g_{bc} \neq 0$ in general but $\nabla_a g_{bc} \equiv 0$ always (for the Levi-Civita connection). The only reason your argument works is because you just did two steps in the wrong order/because your assertion works in local inertial coordinates and this is a legal choice to make. You should write $\partial^b g_{00} = \partial^b (-V^2) = \nabla^b (-V^2)$ but you wrote $\partial^b g_{00} = \nabla^b g_{00} = -\nabla^b (-V^2)$. –  Sharkos Apr 22 '13 at 23:01
    
I get that it has to be coordinate-invariant. I guess what I'm trying to say is that the scalar function, say, $h$, defined in one given coordinate system by $h = g_{00}$, is coordinate-invariant. Clearly under another chart $h$ will in general not still equal $g_{00}$. It is still a scalar. Apologies for not being clearer. –  Stromael Apr 22 '13 at 23:24
    
You can define a scalar function by its value in one coordinate system, of course, yes! But the expression $\nabla_a g_{00}$ means something different to $\nabla h$ even if $h=g_{00}$ in this coordinate system; it contains the partial derivative and Christoffel symbols involving the other components of $g$. –  Sharkos Apr 22 '13 at 23:49
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