Determining the causal structure of spacetime (I)

In this series of blog posts, I aim to explain the causal structure of the spacetime solutions in general relativity. Currently, I am working on a special extension of Einstein’s theory (general relativity) which is known as Horndeski theory. There, I am trying to find the causal structure of the allowed solutions which allegedly permit super luminal propagation of metric perturbations. The methodology to obtain the structure is similar for all the gravitational theories and I wish to demonstrate it for general relativity (which is my comfort zone).

In order to make the observable predictions from a consistent physical theory, we are interested in finding how degrees of the freedom evolve and behave. For that, we try to obtain/formulate equations of motions, which capture the physics at the infinitesimal scale. Once we embed the physics in these equations, we solve them and make the verifiable predictions. For instance, in Newtonian mechanics, we study point particle moving in one dimension $x$ . We, then ask how this degree of freedom behaves or evolves in time (which, again, is an assumption). For such a theory, we have the equation of motion $F=ma=m\frac{d^2 x(t)}{dt^2}$ which contains the spectacular physical insight from Newton (I won’t even try to elaborate that because it deserves a separate blog post). Now this is a linear second order differential equation which requires two initial conditions. In other words, if you give me the initial position and initial velocity, I can tell you the entire future of $x(t)$ by solving the differential equation. In fact, for constant $F$ , we get $x(t)=ut+\frac{1}{2}\frac{F}{m}t^2+x_0$

Another way to look at it: equation(s) of motion (of a theory) give us the prescription to evolve the initial data (known values of the degrees of freedom) to final data that we are interested in.

Now Einstein’s gravity theory has the components of the metric as the degrees of freedom. We denote them by $g_{\mu\nu}$ . Another radical difference (common to all relativistic theories) is that the space and time are unified into a single parameter space called spacetime manifold (inheriting all the structures of the pseudo-Riemannian manifolds). So here we ask our favorite question: how do $g_{\mu\nu}$ evolve in the spacetime. Mathematically, we want to know the $g_{\mu\nu}(x)$ , where $x$ is the representative of the spacetime coordinates from now on.

This time, we use the insight from Einstein to write the equation of motion for general relativity (in vacuum) as

$R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=0=E_{\mu\nu}$

where the symbols have their usual meaning. Here we are working on the 4 dimensional pseudo-Riemannian manifold $(\mathcal{M},g)$ on which $g$ is to be determined by the equation of motion. We define a Cauchy surface $\mathcal{C}$ which is a c0dimension 1 surface in $\mathcal{M}$ on which the initial data for Einstein’s equations is defined. So what does this initial data comprised of? (We will take a small detour and return to the topic in next blog post)

Helmholtz wave

In order to understand that, we start with a simple example. Consider a 2+1 manifold as shown. Now let us consider a wave operator on this manifold given by $\hat{L}=-\frac{\partial^2}{\partial t^2}+\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}$ . For a scalar degree of freedom denoted by $u$ the operator $\hat{L}$ acts on it to generate a wave equation $\hat{L}u=0$ . Here we are not interested in solving the linear second ordered partial differential equation. We want to deduce the properties of the wave, like it’s speed of propagation and the extent to which a disturbance/fluctuation in $u$ can propagate.

Let us say that we know the value of $u=u(x,y)$ on $\phi(x,y,t)=t=0$ surface (where we might have a specified source or some boundary condition). The surface is essentially a codimension-1 surface and we will call the coordinates $x,y$ as internal coordinates (w.r.t the surface). Thus we can now easily compute the internal derivatives (from $u(x,y)$ ) and denote them by $u_i=\partial_iu(x,y)$ where $i\in (x,y)$ . Let the exterior/normal derivative to the surface be $u_\phi (x,y)=\eta(du,d\phi)=-\partial_tu$ (note that $d\phi = dt$ on $\mathcal{C}$ ).

In this example, we have a Cauchy surface $\mathcal{C}$ defined by $\phi =t$ with the data

$u(x,y)$
internal derivative $u_i(x,y)$
external derivate $u_\phi(x,y)$

given on $\mathcal{C}$ . Clearly the second derivatives of $u$ except $u_{\phi\phi}$ can be computed from the given data. And here we use the wave equation (or equation of motion in general) to find the missing second derivative of $u$ . The co-normal to the surface $\phi=0$ is given by $d\phi$ which we can easily represent in the natural co-normal basis as $d\phi=\phi_\mu dx^\mu$ . And then we perform the coordinate transformation to chart $\lambda^\mu$ such that $\lambda^0=\phi$ . Note for this particular case $d\phi \parallel dt$ and the new coordinate chart is exactly equal to $x^\mu$ chart. This is because $\mathcal{C}$ is already perpendicular to the time coordinate.

It is not very difficult to show that the wave equation $\hat{L}u=0$ , under the above transformation, converts into $u_{\phi\phi}Q(\phi_\mu)+\ldots=0$ where $Q(\phi_\mu)=\eta^{\mu\nu}\phi_\mu\phi_\nu$ and we call it the characteristic form. Now there are two situations

$Q\neq 0$ : in this case, we can invert the equation $u_{\phi\phi}Q(\phi_\mu)+\ldots=0$ and find the second derivative of $u$ in the direction normal to the surface. With that information we can easily evolve the data further in time.
$Q=0$ : well, we can’t invert the equation which basically implies that there is no unique evolution of $u$ beyond that surface (which is now called the characteristic surface).

In our example, the characteristic form is identically 1 (just look at the coefficients of second derivatives (-1+1+1). Hence we don’t have any characteristic hypersurface for the degree of freedom $u$ obeying the wave equation $\hat{L}u=0$ with the Cauchy surface as the entire $x,y$ plane. It is not surprising if you think about the plane electromagnetic waves, which again don’t have any characteristic hypersurface and obey the same wave equation we wrote above.

Moving on, we obtained an equation for a surface in new coordinates $\lambda^\mu$ given by $Q(\phi_\mu)=0$ . Physically, this is the surface beyond which we can not evolve the degrees of freedom uniquely (as the second derivative is not uniquely determined). Now we need a mechanism to generate this surface.

Bicharacteristic curves

First we define the bicharacteristic curves or rays which are related to the linear second order partial differential operator $\hat{L}$ , where we now generalize it to $\hat{L}[u]=a^{\mu\nu}u_{\mu\nu}+d$ where $u_{\mu\nu}=\partial_\mu\partial_\nu u$ . It is not difficult to check that in this case the characteristic form becomes $Q=a^{\mu\nu}\phi_\mu\phi_\nu$ . For our Helmholtz wave, we get the form $Q=-\phi_t^2+\phi_x^2+\phi_y^2$ .

Now the bicharacteristic curves are generated in by the linear ordinary differential equations, with a parameter $s$ , given by

$\frac{dx_\mu}{ds}=\frac{1}{2}\partial_{\phi_\mu}Q$ and $\frac{d\phi_\mu}{ds}=-\frac{1}{2}\partial_{x_\mu}Q$

For our Helmholtz wave, one can easily solve them and find the solutions $x(t)=at+b, y(t)=ct+d$ . They actually form the rays of a cone (with appropriate boundary conditions) traveling with speed 1.

We have shown that if we introduce some perturbations at some point in Minkowski manifold, those perturbations will travel at unit speed and won’t escape the cone. This essentially exhibits the causal structure of flat Lorentzian spacetime which is in concurrence with the wave equation.

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Ravi Mohan

If you haven't found something strange during a day, it hasn't been much of a day. – J. A. Wheeler