# Wald GR, Chapter 1 Notes

## 1.2: Space and Time in Prerelativity Physics and in Special Relativity

• Inertial motion is the motion a material body undergoes when it is not subjected to any external forces.

## 1.3: The Spacetime Metric

• In classical physics, invariants between inertial frames are: (1) the time interval $\Delta t$ between two events, (2) the spatial distance $|\Delta x|$ between two simultaneous events.

• In special relativity, the invariant between inertial frames is the interval $$I=(\Delta s)^2=-(\Delta t)^2+(\Delta x)^2+(\Delta y)^2+(\Delta z)^2.$$ The Poincare/Lorentz transforms are those linear transformations which leave the interval unchanged.

• The equivalence principle is that the gravitational force on a body is proportional to its inertial mass (i.e. that gravitational mass, which serves as the coupling to the gravitational force, is the same as inertial mass).

• Mach’s principle: “non-accelerating” and “non-rotating” are meaningless with no reference to objects/matter (i.e. in a vacuum). Allegedly this encouraged Einstein to think about the presence of matter influencing the nature of spacetime.

• GR: “The intrinsic, observer-independent, properties of spacetime are described by a spacetime metric, as in special relativity. However, the spacetime metric need not have the (flat) form it has in special relativity. Indeed, curvature, i.e., the deviation of the spacetime metric from flatness, accounts for the physical effects usually ascribed to a gravitational field. Furthermore, the curvature of spacetime is related to the stress-energy-momentum tensor of the matter in spacetime via an equation postulated by Einstein.”

## Problem 1: Car & Garage Paradox

A car of proper length $L$ is travelling towards a garage, of proper length $L$, at speed $\beta$. There is a man at the entrance of the garage who is going to close the door once the car has entered. The car will happily plow through the back of the garage. The doorman says that the car spends some non-zero amount of time completely contained in the garage after he has closed the door. The driver of the car says that he crashed through the back of the garage before the door was closed.

Let the unprimed coordinates $(t,x)$ be the garage reference frame. If we draw an $xt$-plane, then the garage’s “worldline” is a rectangular strip that goes from some $x=x_0$ to $x=x_0+L$, while extending indefinitely along $t$. It stands there, with that spatial extent, for all time.

On the same graph, we can draw the coordinate axes for the car reference frame. Those coordinates, $(t',x')$ are related to the first by a Lorentz boost, with the familiar relations, e.g. $t'=\gamma(t-\beta x)$. It’s then a simple matter to draw the axes $t'=0$ and $x'=0$: they are lines in the $xt$-plane. Specifically, the $x'$ axis is $t=x/\beta$ and the $t'$ axis is $t=\beta x$; they are a “squished” version of the $x$ and $t$ axes, symmetric about the line $x=t$.

The worldline of the car is simple to describe in the $(t',x')$ coordinates: it’s a strip between $x'=x'_0$ and $x'=x'_0+L$, extending indefinitely in the $t'$ direction. Seen in the $xt$-plane, this rectangular strip is tilted; thus its projection onto the $x$-axis is shortened from its real length (this is length contraction). Note that this tilted region, viewed in the $xt$-plane, implies motion of the car from the garage frame (and vice versa). The plane drawn represents “reality” and we are imposing two arbitrarily chosen coordinates on it.

Now we can examine the situation at a fixed time in either frame of reference. The two rectangles, one straight and one tilted, each There exist times $t$ in the garage frame such that the $x$-extent of the car is fully contained within the $x$-extent of the garage. However, with fixed $t'$, there is no time when the car is fully inside of the garage.