What is the notion of a spatial angle in general relativity?

Question

Is there a notion of spatial angles in general relativity?

Example: The world line of a photon is given by $x^{\mu}(\lambda)$. Suppose it flies into my lab where I have a mirror. I align the mirror in such a way, that I measure a right angle between the incoming and outgoing photon. How can I now calculate the wordlline of the outgoing photon?

Of course I cheated a bit since I just used the old euclidean notion of an angle when I say "I measure a right angle between the incoming and outgoing photon". But I hope that this is allowed locally.

Answer 1

If it is a photon, then you know that $x^{a}x_{a}$ is always zero. You have measured the spatial direction of the photon with your mirror apparatus, so you know the values of the $x^{i}(0)$ at some time${}^{1}$, which we will call zero. Furthermore, we can infer the value of $x^{0}$ at this time from the fact that $x^{a}x_{a} = 0$.

Then, all we need is to know the metric tensor, and we can simply solve the set of ODEs given by,

$$\frac{\mathrm{d}^2 x^a (\lambda)}{\mathrm{d}\lambda^2} + \Gamma^a_{bc} \, \frac{\mathrm{d} x^b(\lambda)}{\mathrm{d}\lambda} \frac{\mathrm{d}x^c (\lambda)}{\mathrm{d}\lambda} = 0$$

for all subsequent times${}^{2}$. Typically, this will have to be done numerically, but there are some known exact solutions for simplified cases. $\lambda$ is the null affine parameter, and the $\Gamma$'s are the Christoffel symbols, which are defined from the metric tensor. For the familiar example of a spherically symmeteric gravitating body AND for weak gravity in nonrelativistic motion, the only relevant component of the metric tensor is $g_{tt} = -\left(1-\frac{2M}{r}\right)$, and the only relevant Christoffel symbol is $\Gamma_{tt}{}^{r} = \frac{M}{r^{2}}$, and our theory becomes Newtonian gravity.

 

 

 

${}^{1}$Here, I take letters from the beginning of the alphabet to be spacetime indices, while letters from the middle of the alphabet are sums over only the spatial components.

${}^{2}$ note that "time" (given by $\lambda$) here doesn't really mean time in the proper sense. Null geodesics don't experience any clock motion. Null geodesics, however, DO trace out "null affine parameter", which you can interpret as how many perpendicular null geodesics they cross as the photon moves.

Answer 2

As angles are a local notion (they're defined on the tangent bundle between two vectors), there aren't a lot of issues to define them in GR, but spatial angles in particular are gonna be defined up to some choice of a spacelike hypersurface, since they must be defined in space, so to speak (you can see the variation of the notion with that choice for instance with the notion of relativistic abberation).

If you have say two curves $\gamma_1, \gamma_2$ meeting at a given point, with tangent vectors $u_1, u_2$ we can define an angle between the two using the metric tensor,

\begin{equation} a(u_1, u_2) = \frac{g(u_1, u_2)}{\|u_1\| \|u_2\|} \end{equation}

which is well defined for every vector except with null vectors, but those will not be spatial angles. To define a spatial angle we need to define some spacelike hypersurface $\Sigma$ going through their point of intersection, in which case this surface induces a subspace $W$ in the tangent bundle, on which you should then project your metric tensor and your vectors.

As usual in the case of GR, give, a small enough neighbourhood, the angles behave much in the same way as they do in Euclidian space, although this is not guaranteed to extend outside of that.

Answer 3

As a geometric quantity, the value of an "angle" can be determined and expressed in a coordinate-free way:

Given three pairwise space-like events, "$A$", "$B$" and "$C$", and given the positive real numbers
$\frac{\sigma[ ~ A C ~ ]}{\sigma[ ~ A B ~ ]}$, $\frac{\sigma[ ~ A C ~ ]}{\sigma[ ~ B C ~ ]}$ and $\frac{\sigma[ ~ B C ~ ]}{\sigma[ ~ A B ~ ]} = \frac{\sigma[ ~ A C ~ ]}{\sigma[ ~ A B ~ ]} / \frac{\sigma[ ~ A C ~ ]}{\sigma[ ~ B C ~ ]}$ as ratios between values of Synge's world function $\sigma$ (or equivalently between signed squares of lengths of arcs consisting of overlapping straight sections) between event pairs,

then the value of "angle at $B$, between $A$ and $C$" may either be expressed (directly) as

$\angle [ ~ A B C ~ ] := $ $\text{ArcSin} \left[ \frac{1}{2} \sqrt{ 2 + 2 \frac{\sigma[ ~ A C ~ ]}{\sigma[ ~ A B ~ ]} + 2 \frac{\sigma[ ~ A C ~ ]}{\sigma[ ~ B C ~ ]} - \frac{\sigma[ ~ B C ~ ]}{\sigma[ ~ A B ~ ]} - \frac{\sigma[ ~ A B ~ ]}{\sigma[ ~ B C ~ ]} - \frac{\sigma[ ~ A C ~ ]}{\sigma[ ~ A B ~ ]} \frac{\sigma[ ~ A C ~ ]}{\sigma[ ~ B C ~ ]} } \right];$

or, more generally, in terms of $\sigma$-values (or of signed square values of lengths of arcs consisting of overlapping straight sections) involving additional events "$F$", "$G$" (named in the sense of variables) which are space-like to each other as well as to "$A$", "$B$" and "$C$", as

$\angle [ ~ A B C ~ ] := \text{Limit}_{\{ F, G \}} {\huge[} $
${\Large \{ } \frac{\sigma[ ~ B F ~ ]}{\sigma[ ~ A B ~ ]} \rightarrow 0, \frac{\sigma[ ~ B G ~ ]}{\sigma[ ~ B C ~ ]} \rightarrow 0, $
$2 + 2 \frac{\sigma[ ~ B F ~ ]}{\sigma[ ~ A B ~ ]} + 2 \frac{\sigma[ ~ B F ~ ]}{\sigma[ ~ A F ~ ]} - \frac{\sigma[ ~ A F ~ ]}{\sigma[ ~ A B ~ ]} - \frac{\sigma[ ~ A B ~ ]}{\sigma[ ~ A F ~ ]} - \frac{\sigma[ ~ B F ~ ]}{\sigma[ ~ A B ~ ]} \frac{\sigma[ ~ B F ~ ]}{\sigma[ ~ A F ~ ]} \rightarrow 0, $
$2 + 2 \frac{\sigma[ ~ B G ~ ]}{\sigma[ ~ B C ~ ]} + 2 \frac{\sigma[ ~ B G ~ ]}{\sigma[ ~ C G ~ ]} - \frac{\sigma[ ~ C G ~ ]}{\sigma[ ~ B C ~ ]} - \frac{\sigma[ ~ B C ~ ]}{\sigma[ ~ C G ~ ]} - \frac{\sigma[ ~ B G ~ ]}{\sigma[ ~ B C ~ ]} \frac{\sigma[ ~ B G ~ ]}{\sigma[ ~ C G ~ ]} \rightarrow 0 {\Large \} }; $
$\text{ArcSin} \left[ \frac{1}{2} \sqrt{ 2 + 2 \frac{\sigma[ ~ F G ~ ]}{\sigma[ ~ B F ~ ]} + 2 \frac{\sigma[ ~ F G ~ ]}{\sigma[ ~ B G ~ ]} - \frac{\sigma[ ~ B G ~ ]}{\sigma[ ~ B F ~ ]} - \frac{\sigma[ ~ B F ~ ]}{\sigma[ ~ B G ~ ]} - \frac{\sigma[ ~ F G ~ ]}{\sigma[ ~ B F ~ ]} \frac{\sigma[ ~ F G ~ ]}{\sigma[ ~ B G ~ ]} } \right] {\huge]} . $

If the region containing the events $A$, $B$ and $C$ is flat then these two "angle" values are equal.