Schutz description of Galilean invariance of interval

Question

In B. Schutz's textbook "A First Course in General Relativity", there is a sentence on page 172 discussing Galilean relativity and how the distance between events is invariant in coordinate systems moving relative to one another. The sentence goes as follows:

"If the events happened at different times, each observer would take the location of the events in their respective space slices and compute the Euclidean distance between them. The locations would differ for different observers, but again the distance between them would be the same for all observers."

I do not understand this sentence. In a frame in which both events occur at the same point in space the distance between them is zero, and in a frame moving relative to the first they will have a non-zero distance between them. What am I missing?

Answer 1

I think you are right, this is an error in Schutz's book.

I think what he wants to do is to draw a parallel to how Lorentz transformations are defined. Lorentz transformations (which are the group of symmetries that define special relativity) are transformations of the coordinates $x^\mu \rightarrow \Lambda^\mu_{\ \ \nu} x^\nu$ (where $x^0=t$ and $x^i$ refer to the $x$, $y$, and $z$ axes in Cartesian coordinates) which leave the spacetime interval $$ \Delta s^2 = -c^2 \Delta t^2 + \Delta x^2 = \Delta x^\mu \eta_{\mu\nu} \Delta x^\nu $$ invariant. In this case, what he says is perfectly true: inertial observers may disagree on the values of the coordinates $x^\mu$, but they will agree on the spacetime interval $\Delta s^2$. However, this statement does not carry over in a straightforward way to Galilean relativity.

We can recover Galilean relativity in the limit $c\rightarrow \infty$. However, if we think of the limit carefully, there are two cases.

• Case 1: $\Delta t=0$. Then the interval reduces to the Euclidean distance.

• Case 2: $\Delta t \neq 0$. If we divide both sides by $c^2$, then the equation becomes $\Delta s^2/c^2 = -\Delta t^2 + \Delta x^2/c^2$. For a fixed, finite spatial distance $\Delta x^2$, that term goes away, and the interval becomes $\Delta s^2/c^2 = - \Delta t^2$.

This shows that you cannot take a simple limit of special relativity, and obtain the case Schutz is discussing.

However, both case 1 and case 2 do describe Galilean relativity. The main point is that there is a universal time $t$ that all observers agree on. Or, at least they can synchronize their clocks so they all use the same time. The two cases above lead to the following two facts, which are true in Galilean relativity:

• Fact 1: When $\Delta t=0$, then all observers agree on the distance between events in their frames. (Since all observers agree on $\Delta s^2$, and when $\Delta t=0$, then $\Delta s^2$ is the squared Euclidan distance).

• Fact 2: When $\Delta t\neq 0$, then all observers agree on the time interval between the events. (Since all observers agree on $\Delta s^2$, and when $c\rightarrow \infty$, $\Delta s^2$ is proportional to $\Delta t^2$).

More formally, I think the problem is that the Lorentz transformations are linear transformations of the spacetime coordinates, but the Galilean transformations are affine transformations of the spatial coordinates, meaning that the spatial coordinates include a shift proportional to $v \Delta t$. Schutz somehow does not seem to have properly accounted for this difference in his discussion.

You could make sense of his statement if you accepted there was some universal rest frame. Relative to a universal rest frame, what he says is true. But that seems contrary to the ideas of relativity.

Answer 2

Limiting ourselves to time t and motion in a single space dimension x, the Lorentz transformations in Special Relativity are:

$$x' = \gamma (x-vt),$$ $$t' = \gamma (t-xt),$$

where the primed coordinates are the those of the moving frame and using units such that c=1.

An object with proper length $L_0 = (x_2'-x_1')$ in the primed frame, can be written in terms of the transformed coordinates as $L_0 = \gamma (x_2 - vt_2) - \gamma(x_1 - vt_1)$.

For a moving object, the only sensible measure of length, is to measure the end points simultaneously.

However, in SR, even if the coordinates of the end points of an object are measured simultaneously in the rest frame of the object, those coordinates are not simultaneous in the frame that sees the object as moving. The time coordinates have to be adjusted so that they are simultaneous and so we set $t_2 = t_1$ and the above equation becomes $L_0 = \gamma(x_2 - x_1)$ and conclude that the distance between the endpoints in the unprimed frame is $(x_2 - x_1) = L/\gamma$. This is the well known length contraction formula. Length is definitely not invariant in SR.

In Galilean coordinates the corresponding space and time transformations are:

$$x' = x-vt$$ $$t' = t$$

Now following the same procedure as above:

$L_0 = (x_2'-x_1') = (x_2 - vt_2) - (x_1 - vt_1) $

As mentioned above, the only sensible measure of length for a moving object, is to measure the end points simultaneously and this applies to equally to Minkowski coordinates or Galilean coordinates, so $t_2$ must be set equal to $t_1$ and the result is:

$L_0 = (x_2'-x_1') = (x_2-x_1) $.

In other words the spatial distance between the endpoints of an object are the same in any reference frame in Galilean coordinates.

Now, I am talking about the simultaneous spatial separation between the endpoints of an object, which we normally call the length of the object. Schutz on the other hand, is talking about the spatial separation of two events. This is problematic, because events are not normally thought of as having a velocity or a rest frame, so it is difficult to justify Schutz's claim that the spatial separation of two events is invariant under a Galilean transformation.

Schutz also explains Galilean spatial separation in terms of space slices. I checked the earlier chapters of the book and he does not appear to define what he means by a space slice in this context and so it is hard to know what he meant. I can can only conclude that Schutz has confused the length of a physical object, with distance between two events and his claim is technically incorrect and should be edited and clarified in any later editions.