Chapter 3

Special Relativity

In Special Relativity (SR), time is no longer independent of space (i.e., it's not universally experienced to pass at the same rate for all observers). The speed of light $c$ is more fundamental than space or time separately — it is the thing that doesn't change. Space and/or time must "adjust" so that everyone agrees on the value of $c$ . Since speed is distance divided by time, time and/or space may need to be compressed in order for this to work, so these concepts are combined into space-time, with time acting as another dimension. Before SR, everyone would agree on the lengths of objects and intervals of time between events, i.e., $\Delta t$ and $\Delta \ell = \sqrt{\Delta x^2 + \Delta y^2 + \Delta z^2}$ would be the same no matter how observers were moving relative to each other.

Because these two are now linked in SR, the thing that everyone agrees on now (i.e., the thing that is invariant) is the "distance" between events $\Delta s$ . This distance is given by

(\Delta s)^2 = -c^2 (\Delta t)^2 + (\Delta \ell)^2\, .

"Time dilation" and "length contraction" happen in reference frames moving (at $v \sim c$ ) with respect to another, "at rest," frame. The moving observer, being in their own "at rest" frame, would consider the other frame to be moving and thus experience these effects. Both observations are totally self-consistent and valid; in SR, there's no true "at rest" frame, just frames moving relative to each other. SR cannot be used to explain how frames achieve relative velocities (via acceleration, since an accelerated reference frame is not described by SR), but recognizing which frame experienced acceleration clears up most paradoxes. The changing of times/lengths is given by the Lorentz factor $\gamma$ :

\gamma = \left(1-\frac{v^2}{c^2}\right)^{-1/2}

General Relativity

In Newtonian gravity, the gravitational force is the sum of the gravitational attraction of all bits of mass (but including direction, which is ignored here out of laziness):

F_g = \sum_i \frac{Gm_1 m_i}{(r_i-r_1)^2}

In GR, there is no longer any such thing as a gravitational force — space-time is warped by mass (and energy), which causes otherwise straight lines to become curved! We'll return to GR in the next chapter.

What does it mean that straight lines are curved? A straight trajectory, in physics, comes about when no net force acts on an object (Newton's first law). Imagine you drew a straight line on a piece of stretchy graph paper, which represents an $x,y$ Cartesian coordinate system. If you stretch that paper over a ball, equivalent to warping space, the straight line would now appear curved if viewed inside the 2D sheet, without knowing the sheet was stretched over a ball. Because the grid has been stretched more some places than others, the path in the 2D sheet looks curved as long as the line doesn't pass right over top the ball (an object following that trajectory would appear to speed up / slow down, so the path is curved in space-time, but not necessarily in space alone). In terms of gravitational fields, Earth orbits the Sun not because the Earth is attracted to the Sun, but because the Sun warps space-time, causing straight lines to run in circles around it.

Warped coordinates in GR can be very complicated, but if we want the curvature to be identical at all points, there are only 3 mathematical forms it can take. This is a consequence of requiring the universe to be homogeneous and isotropic on large scales. If the universe looks the same no matter where you are in it, the same must hold for curvature. We can write the 3D forms of these geometries in spherical coordinates as

(d\ell)^2 = (dr)^2 + S_\kappa(r)^2 (d\Omega)^2

where $(d\Omega)^2 = (d\theta)^2 + (\sin\theta d\phi)^2$ and

S_\kappa(r) = \begin{cases} R\sin(r/R)\, , & \kappa = +1\\ r\, , & \kappa=0\\ R\sinh(r/R)\, , & \kappa = -1 \end{cases}

The variable $\kappa$ simply selects one of these three possible geometries. The flat, Euclidean geometry we're used to is given by $\kappa = 0$ , and $d\ell^2$ takes the recognizable polar form, where a volume element $\Delta V = (\Delta r) (r\Delta \theta) (r\sin\theta \Delta\phi)$ . Positive or negative values of $\kappa$ correspond to positive (spherical) or negative (hyperbolic) curvature, parameterized by the radius of curvature, $R$ . The meaning of $R$ is easier to visualize with positive curvature, which in 2D can be imagined as the surface of a sphere. As you travel in a straight line, eventually you'll go around the entire surface of the sphere (when $\Delta r=2\pi R$ ), ending up right where you started. The same is true in 3D, but your brain has not evolved to imagine things in 4 spatial dimensions, so you can't visualize the 3D case. Negative curvature is described as "saddle-shaped", although this analogy only works for the center point of the saddle — it has been proven that even a 2D hyperbolic surface cannot be represented in 3 dimensions.

Metrics

In SR, or flat space-time, the "distance" between 2 events is given by the Minkowski metric:

ds^2 = -c^2 dt^2 + dr^2 + r^2 d\Omega^2

Light always takes the shortest route between 2 events, i.e., a straight line, which is defined to be $ds = 0$ . In flat space-time, $d\Omega = 0$ (as long as you choose your coordinate system such that 1 event happens at $r=0$ ), so the equation becomes $0=-c^2 dt^2 + dr^2$ , or

\frac{dr}{dt} = \pm c

Unsurprisingly, light travels at $c$ , the speed of light. This metric is valid in any reference frame (where the observer in that frame can define coordinates as if they're at rest, even if they accelerated to change their speed), which is why all observers must measure light to travel at $c$ in their frame.

This metric can be modified to describe a uniformly expanding (or contracting) universe just by adding the scale factor $a(t)$ :

ds^2 = -c^2 dt^2 + a(t)\left[dr^2 + r^2 d\Omega^2\right]

This modification is possible because the spatial dimension of the universe changes the same way everywhere at the same time. Even though in SR (and GR) there is no preferred reference frame, there is a reference frame of greatest symmetry, which can be thought of as the universe's "at rest" reference frame.

Proper Distance

With this metric, we can now define an instantaneous distance between 2 points at one instant $t$ (this is possible because the universal "at rest" reference frame is the same everywhere, with time running at the same rate in this frame). This distance is called the proper distance and can be determined by setting $dt = 0$ and integrating $ds$ between 2 points. Again, we define coordinates so $d\Omega = 0$ :

d_p(t) = \int_0^s ds^\prime = a(t) \int_0^r dr^\prime = a(t) r\, .

This recovers our earlier result that $d(t) = a(t) d(t_0)$ , where $r = d(t_0)$ , or the distance between two objects today. We call $r$ a "co-moving" coordinate. It doesn't have to be defined as the distance between two objects today (could be defined as the distance 1 billion years, or 1 Gyr, after the Big Bang, for example), but we will define it this way. If two galaxies are not moving relative to the "background flow" of the universe (its "at rest" frame), their co-moving distance will always be the same, even though their true, physical distance when $a = \frac{1}{2}$ is half what it is today, when $a=1$ .

Taking the derivative of the proper time, we find

\dot{d}_p(t) = \dot{a} r = \frac{\dot{a}}{a} d_p(t) = v

Remembering Hubble's law, $v = H_0 d$ , we can define Hubble's constant from this relation as

H_0 = \left( \frac{\dot{a}}{a} \right)_{t=t_0}\, .

Similarly, at some time in the past, we have $H(t) = \dot{a}/a$ for $a$ and $\dot{a}$ at that time $t$ (in general, $H(t) \ne H_0$ except at $t=t_0$ ). Because it is only constant at the current time, the more generic term for this variable is the Hubble parameter.

Using proper distance, we can figure out how the expansion of the universe affects the wavelength of a photon. Basically, you follow 2 consecutive crests of a wave, calculating the proper distances at the different times the crests were emitted, making some reasonable assumptions, and expressing wavelengths in terms of redshift. The derivation is in the textbook (and we'll do it in class), but you end up with the important result

1+z = \frac{\lambda_o - \lambda_e}{\lambda_e} = \frac{a(t_0)}{a(t_e)} = \frac{1}{a}\, .

This uses our usual definition of $a(t_0) \equiv 1$ and $\lambda_o$ is the observed wavelength at the current time $t = t_0$ . Once again, this time with feeling:

\boxed{\boxed{1+z = \frac{1}{a}}}

So, if you observe a galaxy that has a redshift $z = 1$ , you don't necessarily know how far back in time you're observing the galaxy, but you DO know that the universe was 1/8th as large in volume since $a = 1/2$ .