Metric of spacetime

When we defined the inner product of 4-vectors we stated that

\begin{align*} \fv x \cdot \fv y = x^\alpha y_\alpha = -x^0 y^0 + x^1 y^1 + x^2 y^2 + x^3 y^3. \end{align*}

This is a little bit of a strange requirement. We haven’t yet shown how $y_\alpha$ is related to $y^\alpha$ or where the negative sign comes from. The metric of spacetime formalizes this.

We define the Minkowski metric of spacetime (in cartesian coordinates) as:

\begin{align*} \eta_{\alpha\beta} = \mat{-1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1}. \end{align*}

Then we can write the inner product above as $\fv x \cdot \fv y = \eta_{\alpha\beta} x^\alpha y^\beta = x_\beta y^\beta$ . We see the negative appears from the $\eta_{00}$ component. We can think of $\eta_{\alpha\beta} x^\alpha$ as “lowering” the index of the $x$ , transforming it from an up vector to a down vector (or from a vector to a 1-form).

Since the vectors can be decomposed in terms of their unit vectors, $\fv x = x^\alpha \fv e_\alpha$ , we should be able to equivalently write $\fv x \cdot \fv y = (x^\alpha y^\beta)(\fv e_\alpha \cdot \fv e_\beta)$ . By our definition above, we can say $\fv e_\alpha \cdot \fv e_\beta = \eta_{\alpha\beta}$ . Finally, this defines why $\fv e_0 \cdot \fv e_0 = -1$ .

In other coordinate systems and in general relativity, the metric of spacetime will be more complicated. In those cases, we will write it as $g_{\alpha\beta}$ . We use $\eta_{\alpha\beta}$ to refer to this specific metric.

We define an inverse metric $\eta^{\alpha\beta}$ by requiring that $\eta_{\alpha\beta} \eta^{\alpha\beta} = \delta^\alpha{}_\beta$ , where $\delta^\alpha{}_\beta$ is the Kronecker delta, or equivalently the identity matrix. Using this requirement we show that the inverse metric can be used to raise an index:

\begin{align*} \eta^{\alpha\beta} x_\alpha &= \eta^{\alpha\beta} (\eta_{\alpha\mu} x^\mu) = (\eta^{\alpha\beta} \eta_{\alpha\mu}) x^\mu \\ &= \delta^\beta{}_\mu x^\mu = x^\beta. \end{align*}

Component-wise, $\eta^{\alpha\beta}$ is identical to $\eta_{\alpha\beta}$ .

We’ve seen how up-vectors transform following the Lorentz transform. Let’s use our new metric to consider how down-vectors transform.

\begin{align*} a_{\alpha'} &= \eta_{\alpha'\beta'} a^{\beta'} \\ &= \big( \Lambda^\mu{}_{\alpha'} \Lambda^\nu{}_{\beta'} \eta_{\mu\nu} \big) \big( \Lambda^{\beta'}{}_\sigma a^\sigma \big) \\ &= \big( \Lambda^\mu{}_{\alpha'} \Lambda^\nu{}_{\beta'} \Lambda^{\beta'}{}_\sigma \big) \eta_{\mu\nu} a^\sigma \\ &= \Lambda^\mu{}_{\alpha'} \delta^\nu{}_\sigma \eta_{\mu\nu} a^\sigma \tag{✨} \\ &= \Lambda^\mu{}_{\alpha'} \eta_{\mu\nu} a^\nu \\ &= \Lambda^\mu{}_{\alpha'} a_\mu. \end{align*}

In line $(✨)$ we take advantage of the fact that a transformation of a coordinate $\sigma$ to the primed frame coordiante $\beta'$ and back to the unprimed coordinate $\nu$ results in the same starting coordinate, ensuring $\sigma = \nu$ .

We see that to transform a down-vector we line up its indices with the Lorentz matrix. This should be intuitive.

Weak gravity metric

The Minkowsky metric doesn’t describe general relativity well. In weak gravity we can use a metric that’s not too different, though. “Weak” gravity describes almost everything in our solar system.

The weak gravity metric is

g_{00} = -(1 + 2\Phi), \quad g_{11} = g_{22} = g_{33} = (1-2\Phi).

Here $\Phi = -GM/r$ is the Newtonian gravitational potential. This spacetime describes the weak gravity ( $\Phi \ll 1$ ) around a point mass at $r=0$ with mass $M$ .

Schwarzschild metric

Karl Schwarzschild found the first exact solution. It describes the spacetime outside a spherical non-rotating body of mass $M$ .

g_{00} = -\left(1 - \frac{2GM}{r}\right), \quad g_{11} = \left(1-\frac{2GM}{r}\right)^{-1}, \quad g_{22} = r^2, \quad g_{33} = r^2 \sin^2 \theta.

This solution is exact, although the situation it describes is uncommon — almost everything has angular momentum in reality.

Schwarzschild is a particularly simple strong gravity solution. It also has several nice invariants, which we find by applying the Lagrangian. In particular,

\begin{align*} \hat E &= \left(1-\frac{2GM}{r}\right) \frac{\dt}{\d\tau}, \\ \text{and } \hat L_z &= r^2 \frac{\d\phi}{\d\tau} \end{align*}

are conserved for motion confined to the $\theta=\pi/2$ plane. These are energy and angular momentum per mass.

Kerr metric

The Kerr metric describes a spherical black hole with angular momentum $J = aM$ about $\theta = 0$ . It is a glorious mess.

\begin{align*} g_{00} &= - \frac{\Delta - a^2 \sin^2 \theta}{\Sigma}, g_{11} = \frac\Sigma\Delta, \quad g_{22} = \Sigma, \\ g_{33} &= \left( \frac{(r^2 + a^2)^2 - a^2 \Delta \sin^2 \theta}{\Sigma} \right) \sin^2 \theta, \\ g_{03} &= g_{30} = -\frac{2a \tilde M r}{\Sigma} \sin^2 \theta. \end{align*}

Where

\Delta = r^2 - 2\tilde M r + a^2, \quad \Sigma = r^2 a^2 \cos^2 \theta, \quad \tilde M = GM, \quad a = \frac Jm.