Cauchy problems in general relativity

The Einstein field equation

G_{\alpha\beta}=T_{\alpha\beta}

, as we have said, is highly non-linear. One other problem is, how to construct a time-space with some matter distribution on it such that the above equation is satisfied. This is not an easy problem. Note that there is a term ‘time’ in the equation, this indicates us that perhaps we can start from some low-dimension manifold(hyper-surface) and consider the evolution problem. This is the main idea for the Cauchy problem in general relativity.

Here for simplicity, we consider the case where there is no matter in the time-space, that is to say, T=0.

In general situation for the Cauchy problem, we seek some equation, say, f:\mathbb{R}^{1+3}\rightarrow \mathbb{R} such that Pf(t,x)=0,f(0,x)=f_0(x),\partial_tf(0,x)=f_1(x) where Pf just represents an operator. What about the situation for a time-space? Here the f should be the Riemann metric g, or something equivalent, operator P correspond to the Einstein tensor, that is G(g). What about the initial conditions? Since the starting point is a hyper-surface, this hyper-surface itself has a Riemann metric, g', so g(0,x)=g'(x) corresponds to f(0,x)=f_0(x). As for the second condition, note that \partial_t in some sense here means the normal vector for this hyper-surface. This reminds us to consider the second fundamental form k which involves the normal vector field. So, the second fundamental form corresponds to the condition that \partial_tf(0,x)=f_1(x).

In mathematics, we use the Ricci tensor instead of the Einstein tensor to express the Einstein field equation. So, in the vacuum case, the field equation becomes

Ric_{\alpha\beta}=0

Yet we can show that, if we pose the condition that Ric_{ij}=0 for i,j=1,2,3 on the whole time-space Mand the initial condition Ric_{0\alpha}=0 on the hyper-surface \Sigma, then Ric_{0\alpha}=0 on all the M.

This result, in some sense, resembles exactly to the Cauchy problem in general.

Next I will try to count the degree of liberties. Note that, the differential equation Ric_{ij}(g)=0 is of second order. So, to have exactly finitely many possible solutions, we should set the initial values of g and its first derivative with respect to time \partial_tg(thus on \Sigma). In the above, in fact we just set six set of values, that is, g'_{ij} and k_{ij}. But g has ten independent components, which means that to fully resolve the differential equation Ric_{\alpha\beta}(g)=0(note here I write deliberately the greek letters as indice, which range 0,1,2,3, to indicate that the full Einstein equation in the vacuum is the same as Ric_{\alpha\beta}). That is why we pose directly the other four condition as Ric_{0\alpha}=0 on the whole M. Thus, in this way, we can say that this differential equation with the initial conditions are exactly-determined, or admits finitely many solutions.

Can we say something more about the nature of Einstein field equation? In fact, we can show that, yet under some conditions on the choice of coordinates.

Here is one example. We say that a local coordinate (t=x^0,x^1,x^2,x^3) is a wave coordinate if each axis satisfies \Box x^{\alpha}=0. The operator \Box is the d’Alembertian(in Riemannian manifold, we replace this by the Laplace operator, \Delta x^{\alpha}=0). We can show, that under this kind of coordinate, the Ricci tensor has a particular expression

Ric_{\alpha\beta}=-1/2g^{\gamma\nu}\frac{\partial^2g_{\alpha\beta}}{\partial x^{\gamma}\partial x^{\nu}}+F(g)(\partial g)^2

where F(g) is a function on g and (\partial g)^2 is a quadratic form on the first derivatives of g. So, we see easily, that essentially, the Einstein field equation in the vacuum is a wave equation, or a hyperbolic differential equation.

Yet this kind of choice(which is generally called, the gauge) is a bit special, if we choose another type of coordinate, then the nature of the differential equation will change, too.

Meanwhile, a choice of coordinate is necessary, in that the Einstein field equation is invariant under diffeomorphisms. So in order to choose a representative from the isomorphism class, we need to pose conditions on the coordinates.

One last remark. A priori, we can pose conditions on the coordinates only on the initial manifold, \Sigma. So, there is a problem of coherence. That is to say, if we pose some condition about the coordinates on \Sigma, then we should verify that this condition is also satisfied on the whole M.

There is a very nice review on the Cauchy problem in general relativity, see here. The chapter 7 of the book ‘the large scale structure of space-time’ of Hawking and Ellis discusses this problem, too.

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