The spacetime metric of the universe

The Robertson–Walker metric of space covers all three geometries at once: $k > 0$ is closed, $k < 0$ open, $k = 0$ flat. Isotropy about the origin is clear from the angular part, which is the metric of a sphere of radius $a(t)\,r$. Once homogeneity is shown, isotropy about the origin means isotropy about every point. Homogeneity is harder to see, because the origin looks special.

For the closed case Guth sketches an algebraic proof (slide 3): map any point $(r_0, \theta_0, \phi_0)$ to the origin by going to the four Euclidean coordinates $(x, y, z, w)$ of the embedding space, rotating there, and coming back with the same formulas. Rotations preserve the four-dimensional distances, so the new coordinates have the same metric. The result is a set of algebraic formulas for $(r', \theta', \phi')$ in terms of $(r, \theta, \phi)$. Algebraic identities that hold for positive $k$ also hold for negative $k$, so, Guth argues, the same map proves that the open universe is homogeneous too. He does not write it out, since the algebra gets messy.

Three further facts are recalled (slide 4); only the second was checked by a calculation, in Lecture 12:

• Every homogeneous, isotropic three-dimensional space can be given coordinates in which its metric is the Robertson–Walker metric. Other coordinates are possible, but nothing new can appear.
• For $k > 0$ the universe is finite; for $k \le 0$ it is infinite, since the distance from the origin grows without limit as $r$ grows.
• The two-dimensional open universe is exactly the Gauss–Bolyai–Lobachevsky geometry (an optional homework problem maps Klein's coordinates onto Robertson–Walker ones).

Lecture 12 ended with the interval of special relativity. Shrunk to two events infinitesimally close together, it becomes the metric of special relativity, the Minkowski metric, which Guth writes on the board:

General relativity keeps Gauss's idea that $ds^2$ is a quadratic expression in the coordinate differentials. But the coordinates themselves change meaning:

• In special relativity, the coordinates of an inertial frame are distances and times. You hardly need the metric.
• In general relativity, coordinates are arbitrary labels for points of spacetime. No coordinate system on a curved spacetime can measure distances directly, so only the metric turns coordinate differences into physical distances and times.

So there is no privileged coordinate system, and any change of coordinates is allowed. The rule is that the metric is always changed along with the coordinates, so that $ds^2$ between any two nearby events keeps its value: $ds^2$ is coordinate-invariant (transcript p. 9).

In special relativity, $ds^2$ is defined by what inertial observers measure: observers with no forces on them. In general relativity that definition fails, because gravity is not treated as a force like the others. Non-gravitational forces (electric forces, say) are handled much as in special relativity, but gravity becomes part of the geometry of spacetime.

The expanding universe already shows the problem. A comoving observer feels no force at all: as far as she is concerned she is standing still. Yet she sees every other comoving observer accelerating relative to her, as the expansion rate changes (in the matter-dominated models derived so far, it slows down). So she says gravity acts on everyone except herself — and each of them says the same. Whether a gravitational force acts depends on the frame (transcript pp. 10–11).

The classic illustration, which also motivated Einstein, is an elevator (Ryden's textbook discusses it too).

• Cut the cable. If the elevator falls freely, everything inside falls with the same acceleration. A passenger can lift his feet and hover; groceries he lets go of float in front of him. Inside, there is no sign of gravity — although from the ground we would say gravity is clearly acting.
• Rocket in empty space. Now let the elevator stand on the ground floor: the passenger is pressed to the floor and must hold up his groceries. Then move the same elevator to empty space and tow it with a rocket accelerating at $g$. Inside, he feels exactly the same.

In both pairs, one situation has gravity and the other has none, yet the passenger cannot tell which one he is in. This is the equivalence principle: an accelerating frame without gravity is physically equivalent to a gravitational field. Everything gravity does to a physical system can therefore be found by working out what acceleration does to it.

The special class of observers can now be named. It cannot be "no forces at all", since whether gravity acts depends on the frame. Instead it is observers with no non-gravitational forces acting on them — like the comoving observers of our universe. They are called free-falling observers: free fall in the everyday sense and simply floating in empty space cannot be told apart.

With that, $ds^2$ has the same meaning as in special relativity, with inertial observers replaced by local free-falling ones (transcript pp. 13–14; Lecture 14 slide 1 summarises it):

• $ds^2 > 0$: there are free-falling observers for whom the two events happen at the same time; $ds^2$ is the square of the distance they measure between them.
• $ds^2 < 0$: there are free-falling observers for whom the two events happen at the same place; $ds^2$ is $-c^2$ times the square of the time between them on their clocks.
• $ds^2 = 0$: a light pulse can travel from one event to the other.

Guth writes down the answer first and then justifies it. The spacetime metric of a homogeneous, isotropic universe is the spatial metric with one extra term:

Two special cases check the terms that are there.

Case 1: $dt = 0$. Only spatial separations remain. These must be described by the Robertson–Walker spatial metric, because that is the most general description of a homogeneous, isotropic space.

Case 2: $dr = d\theta = d\phi = 0$. This is a comoving observer, sitting at fixed coordinates while time passes. The metric gives

so the proper time on her wristwatch equals the change in $t$. That is exactly what we want, because cosmic time was defined as the time on the wristwatches of comoving observers. A coefficient other than $c^2$ would make the watches and cosmic time tick at different rates.

A difficult roommate could ask: what about terms mixing time and space, $dt\,dr$, $dt\,d\theta$ or $dt\,d\phi$? "It wasn't in the book" is not a good answer. The real reason is that such a term would break isotropy: $dr$ changes sign when you reverse direction, so a $dt\,dr$ term treats "outward" and "inward" differently.

Guth makes this concrete with a thought experiment (pp. 16–18). Tweedledee and Tweedledum stand at a point with $r \ne 0$ and synchronise their watches with cosmic time. One sets off towards larger $r$, the other towards smaller $r$, both at the same coordinate speed $v = |dr/dt|$. Each stops when the local cosmic-time clocks read an agreed final time $t_f$, and reads his own watch.

With a $dt\,dr$ term, $dr = +v\,dt$ for one and $-v\,dt$ for the other, so the term adds to $ds^2$ for one and subtracts for the other. Their watches disagree, and the two directions could be told apart by measuring time dilation. In an isotropic universe that must be impossible, so the $dt\,dr$ term is not allowed. The same argument removes $dt\,d\theta$ and $dt\,d\phi$.

That justifies every term. The Robertson–Walker spacetime metric is the complete metric of a homogeneous, isotropic universe, and the rest of the course uses it.

• Any homogeneous, isotropic space can be written with the Robertson–Walker metric; for $k > 0$ it is finite, for $k \le 0$ infinite.
• In general relativity coordinates are arbitrary labels; the metric changes with them so that $ds^2$ is coordinate-invariant.
• Whether gravity acts depends on the frame (equivalence principle, valid in small regions). The special observers are free-falling ones, with no non-gravitational forces acting.
• $ds^2$ means the same as in special relativity, measured by local free-falling observers.
• The spacetime metric is $ds^2 = -c^2dt^2 + a^2(t)\left[\frac{dr^2}{1-kr^2} + r^2(d\theta^2 + \sin^2\theta\,d\phi^2)\right]$. The $-c^2dt^2$ term makes comoving watches read cosmic time; isotropy forbids $dt\,dr$, $dt\,d\theta$ and $dt\,d\phi$ terms.