Chapter 7 Questions

"Different" velocities mean different frames. That includes the case where they have the same speed but are moving in different directions. In special relativity if a frame is noninertial it must be accelerating; it's either moving at constant velocity or it's accelerating. Two noninertial frames will also be accelerating with respect to each other, unless they are accelerating in exactly the same way (i.e. two observers could be in the same accelerating frame at different locations and at rest with respect to each other). (The situation is a bit more complicated in general relativity because inertial reference frames can generally be defined only locally.)

Velocities don't add up like they do in Newtonian mechanics. The relativity of space and time intervals extends to velocity. Equation 7.8 describes how to add two velocities together. For your example we have (.75+.75)/(1+.75*.75) = 1.5/1.5625 = 0.96.

One can prove this relation by examining the Lorentz transform of velocity = dx/dt. This involves some mildly complicated algebra.

The second postulate of relativity requires the speed of light to be the same in all reference frames. Since speed is a change of distance divided by a change of time, if time changes (time-dilation) then length must change too to maintain the constancy of light.
A question sometimes arises: is this contraction real or just an optical illusion of some sort? The answer it is real in every sense. A measurement of something moving will be shorter than of that same object at rest according to every possible test.

The spacetime diagram can provide a graphical illustration of length contraction, i.e. you can see how a standard length is shorter in other frames compared to its rest frame.

Nobody has a problem with realizing that two different reference frames moving with respect to each other don't regard two events at different times as occurring at the same point in space. In special relativity we find that two different observers also don't regard two events separated in space as happening at the same time.

We are used to thinking of time unfolding, and time being the same for all possible observers. This isn't the way it is in SR. Simultaneity is defined as "things happening at the same time." We might also define it in this way: Two events (two points in spacetime) are simultaneous in a given frame if a light signal emitted midway between the spatial locations of those two events arrives at those events. An example would be a lightbulb flashing at the center of a train car. In the train-car's frame the light hits the front and the back of the car at the same time, hence those two events are simultaneous. But to an observer watching the train car go by at some speed the two events cannot be simultaneous because the rear of the train comes forward to meet the light signal while the front of the train moves ahead of the light signal (put another way, the flashing of the light doesn't occur midway between the two events, according to the ground-based observer, because the train is in motion).
Because events that are simultaneous must be spacelike separated, they can have no causal relationship; hence, in a sense, simultaneity is a convention. When we say that a set of clocks is synchronized we are making a statement about how the clocks are set relative to one another in a certain frame. In another frame the clocks will show different times, although all the clocks in a given inertial frame will run at the same rate.

The first thing to understand is that time intervals and space intervals are relative, i.e., the value an inertial observer would get for a measurement of either of them depends on the observer's frame. This is different from what we are used to thinking. Space intervals and time intervals vary between frames according to the Lorentz transformation formula (the gamma or boost factor).

If you were to take the definition for the space-time interval, plug in the lorentz transformation terms as appropriate to change delta t and delta x to a different frame, you would find that the terms cancel in such as way as to leave delta s unchanged, despite the transformation to a different frame. This means that all observers, no matter what their relative motion, will measure the same delta s even though their individual delta t and delta x intervals will be different. The agreement between all the inertial observers means that the quantity is invariant.

But what is delta s, the space time interval? In a mathematical sense, it is just a thing constructed in such a way that the Lorentz tranformation terms cancel out, that is it is designed to be an invariant quantity. Does it have a physical interpretation? It is a measure of a "distance" in spacetime. In ordinary space, a distance can be defined that is invariant under the Galilean transformation: the familiar Pythagorean rule is one such mathematical rule that gives an invariant distance even though separate observers' definition of x and y might be different. The spacetime interval extends this concept to spacetime.
There is one useful interpretation of the spacetime interval. The spacetime interval measured along the timelike worldline of an observer is equal to the proper time of that observer, i.e., what that observer's clock records. Since the spacetime interval is invariant , all observers will agree on the proper time intervals along a worldline, even though they may disagree upon the intervals of space and time along that worldline.

Since proper time is a distance in a geometry, it differs along different paths in that geometry, just as the spatial distance along a path through space depends upon the path. Consider two events in spacetime, separated by a timelike interval, and suppose that several clocks travel along separate worldlines between those two events. Each clock will measure a different proper time along its worldline, but the clock which travels along an inertial path (a straight line in spacetime) will register the longest elapsed proper time. This property of spacetime is in some respects opposite to the familiar Euclidean geometry, in which the straight line is the shortest distance between two points.
In general relativity we will invert this relationship and use the maximization of the spacetime interval to define inertial motion. Spacetimes in general relativity are curved, which means that inertial motion is not necessarily linear in the special-relativistic sense, but we can still use the maximization of proper time interval to determine which path through the spacetime represents inertial motion. But the details of this is a topic for general relativity.

Yes. Density is mass per unit volume, so if a block of some density is moving past you, length contraction would decrease the volume while the mass would increase by the relativistic equation.

Speeds are indeed relative, so you must compute the change in speed from the frame of the spaceship. In this case you wish to accelerate from 0.7c to 0.8c, both measured with respect to the Earth. By the relativistic velocity addition law, the velocity change is not 0.1c in the frame of the spaceship, but is 0.23c. In the spaceship frame, the energy required is given by the corresponding boost factor, which in this case is 1.028. The additional energy needed is thus 0.028mc². The change in boost factor as measured from the Earth frame is Gamma(.8c) = 1.67 - Gamma(.7c)=1.40; the difference is 0.27, so the additional energy required will be 0.27mc², as measured in Earth's frame. Why the difference? The straightforward answer is that a measurement of energy is relative. However, in either frame the amount of energy required to accelerate by the same factor will increase as the ship's speed increases, because of the dependence of the boost factor upon the square of the speed. Accelerating from 0.8c to 0.9c in the Earth's frame corresponds to an increase of 0.35c in the spaceship's frame. In the spaceship frame, the energy required for the increase is 0.06mc², while in the Earth frame it is 0.68mc²; in both cases substantially more energy is required to go from 0.8c to 0.9c than to go from 0.7c to 0.8c.

Copyright © 2005 John F. Hawley