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Summary

Now that we have developed the mathematics necessary for quantifying spacetime, let us return to the conceptual development of spacetime itself. You will recall that spacetime is a four-dimensional reality consisting of three space-like dimensions and one time-like dimension.

The points in spacetime are called events, so spacetime may be conceptualized as event space - the four-dimensional space containing every event that ever happened or ever will happen. Events are one of the important aspects of reality that are completely independent of the observer. In other words, events are invariant.

The distance between two events in spacetime is called the interval. If the two events are separated more in space than in time, the interval between them is said to be spacelike. If they are separated more in time than space, their interval is timelike. And if they are equally separated in space and time, they are said to be null separated. The interval between two events is also an invariant, completely independent of the observer who measures it.

So spacetime is a four-dimensional space filled with invariant events separated by invariant intervals.

Coordinates

In order to identify events and specify their location in spacetime, one uses coordinates. Since spacetime is four dimensional, four coordinates are generally necessary. In other words, spacetime can be spanned by four coordinates, three of which are space-like and one of which is time-like.

The choice of coordinates used to span spacetime is completely arbitrary. But a particular observer in a given inertial frame of reference S will typically use a normal rectangular coordinate system x, y, z, t. A second observer in a second inertial frame S’ moving with respect to the first will typically use a different rectangular coordinate system x', y', z', t'.

It should be obvious that the coordinates used to specify a given event are not invariant even though the event they specify is invariant. Coordinates are relative quantities, clearly depending upon the observer’s frame of reference, his choice of origin, and his choice of directions within that frame of reference.

Spacetime Diagrams

Spacetime diagrams are maps of spacetime constructed by plotting the spatial coordinates of the important events versus time. Since it is impossible to plot four-dimensional spacetime diagrams, and quite difficult to construct three-dimensional spacetime diagrams, most spacetime diagrams are two-dimensional with the space x-axis plotted horizontally and the time t-axis plotted vertically.

This t versus x plot (for a single particle) is exactly the same as the usual x versus t graph used in elementary physics to analyze the motion of a particle except for two things: (1) the axes are reversed, and (2) the graph is considered to be an actual map of spacetime rather than just a plot of two variables on a sheet of paper.

The scale of the diagram is usually chosen so that space and time are plotted in the same units of measurements rather than conventional units. Sometimes the unit of conversion (c = speed of light) is indicated directly on the diagram by plotting ct versus x rather than t versus x. If this ever happens to you, don’t let it confuse you. The conversion factor c is just being used to convert conventional units of time into conventional units of distance so that the vertical scale of the diagram is the same as the horizontal scale.

Events are plotted as points on a spacetime diagram according to their specific x-, t-coordinates.

Physical objects existing in time are plotted as lines. Inertial objects (objects free from any external forces) will plot as straight lines. Accelerating objects will plot as curved lines. Nevertheless, even accelerated objects remain everywhere timelike (always having slopes greater than 45^o on the diagram). This is because physical objects always encounter events separated from one another in time. In other words, in every frame of reference, the events along the worldlines of physical objects are separated more in time than they are in space.

Space (in the usual sense of the word) includes all the events in spacetime that occur at a given instant. In other words, space is that region of spacetime perpendicular to a given time line at a given event. Or in still other words, space is parallel to the x-, y-z-axes at a given instant. On a 2-D spacetime diagram, a rest observer’s concept of space is any line parallel to the x-axis and a moving observer’s concept of space is any line parallel to the x'-axis.

Time, of course, is always parallel to an observer’s worldline. So both space and time are relative concepts, describing different directions in spacetime.

Simultaneity and Time Travel

Two events are said to be simultaneous if they occur at the same time. In other words, two events are simultaneous if they have the same value for the coordinate used to specify time. Two events are simultaneous in the rest frame if they occur at the same value of t. They are simultaneous in the moving frame if they have the same value of t'. Since t and t' are different, simultaneity is different for the two frames of references.

Therefore, if a person, object, or even information could jump instantaneously (relative to one observer) from the earth to a distant planet, then according to some other observer that person, object, or information would arrive at the planet before it ever left the earth. In other words, time travel would be possible and the principle of causality would be violated.

The principle of causality asserts that cause always precedes effect. In other words, the consequence of an action cannot precede the action. You cannot receive a message before it is sent.

Without this principle our universe becomes philosophically inconsistent. For example, if it is possible for you to receive a message before it is sent, what would happen if something later on prevented the message from ever being sent. (Suppose the sender decides not to send it after learning that you already received it.)

Regions of Spacetime

The 3-D spacetime diagram shown below illustrates the worldline of an inertial observer (straight line) at rest with an arrow head pointing in the direction of spacetime which he calls time. The region of spacetime perpendicular to this worldline at an arbitrary event is what this observer calls space. Everything in the space plane is simultaneous with the specified event and occurs in the observer's relative present. Every event above the relative present is in the observer’s relative future. And everything below the relative present is in the observer’s relative past. All of the light rays leaving the specified event travel into the future and form a future light cone. All the light rays arriving at the event come from the past and form a past light cone. Everything above the future light cone is in the absolute future, timelike separated from the event. Everything below the past light cone is in the absolute past, timelike separated from the event. Everything between the future and past light cones is absolutely elsewhere, spacelike separated from the event.

Metric Equation

The distance between two points in n-dimensional space is related to the coordinates specifying those points through an equation called the metric equation. In normal 3-D Euclidean space this equation is also called the Pythagorean theorem:

In 4-D Minkowski spacetime this distance is called the interval between events and the metric equation (in conventional units) becomes:

In 2-D Minkowski spacetime this reduces to:

Of course, any equation designed for conventional units may be converted to one designed for normalized units by setting c = 1.

Lorentz Transformation

The Lorentz Transformation equations are used to convert the coordinates of one inertial frame into the coordinates of another. In general these equations can be quite complicated, but they can be simplified considerably by assuming a standard configuration where (1) the origin of the two coordinate systems are coincident at time t = 0 and at time t' = 0. (2) the x-, y-, and z-axes are spatially parallel to the x'-, y-', z'-axes, respectively, (3) the x, y, z, t coordinate frame is at rest, and (4) the origin of the x', y', z', t' coordinate frame is moving along the x-axis with a speed v relative to the rest frame. In this simplified case, the Lorentz Transformation equations and their inverses in conventional units become.

Lorentz Transform

Inverse Lorentz Transform

Time Dilation

By definition, proper time is the invariant interval between two timelike separated events. It is equivalent to the time measured along the worldline of a particular observer. Relative time, on the other hand, is the projection of proper time along the time line of some other observer. In the figure below, t' is the proper time (measured along the worldline of an observer moving with velocity v in the x-direction) and t is the relative time (projection of t' along the time axis of a rest observer). From the triangle one sees that

This same result could have been obtained by substituting x’ = 0 into the Lorentz Transformation equations. The justification for this substitution is that proper time is measured along the worldline x’ = 0 of the moving observer.

This equation is called the time dilation equation and shows that more time passes for the rest observer than for the moving clock. In other words, a moving clock runs slow by a factor of g.

Length Contraction

The proper length of a an object is its length as measured by someone moving with the object. Its relative length is distance through the worldline of the object in another frame of reference. In the figure at right x' is the proper length of a moving rod (the distance through the rod perpendicular to its worldline t') and x is relative length (the distance through the rod perpendicular to t). From the triangle one sees that

This same result could have been obtained by substituting t = 0 into the Lorentz Transformation equations. The justification for this substitution is that both ends of the moving rod are to be measured at the same time t = 0 in the rest frame.

This equation is called the length contraction equation. In words, it is often expresses as, “A moving rod contracts by a factor of gamma.”

Notice that this contraction is along the direction of motion only. There is no contraction along the y- and z-axes, perpendicular to the direction of motion.

Mass, Energy, and Momentum

In relativistic physics, mass is a four-dimensional vector directed forward in time along the worldline of a material object. The scalar magnitude of this 4-vector is the proper mass m_o of the object. The time component of this 4-vector is the total energy E of the object. And the space component of this 4-vector is the momentum p of the object. In conventional units these three quantities are related through the equation

The laws of conservation of energy and momentum follow from the properties of 4-vector addition. The vector sum of the masses before and after particle interactions must be the same.

Exercises:

E1 Events and Intervals

Click above link to go to exercise 1.

E2 Worldlines

Click above link to go to exercise 2.

E3 Twin Paradox

Click above link to go to exercise 3.

E3 Spacetime Protractors and Rulers

Click above link to go to exercise 4.

Textbook Problems:

Textbook Problems

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ModPhy1/Unit1/SpecialRelativity/