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## Sample Problems and Solutions

We have used the postulates of relativity to examine, in particular examples, how observers in different frames of reference measure different values for lengths and the time intervals. We can gain further insight into how the postulates of relativity change the Newtonian view of time and space by examining the transformation equations that give the space and time coordinates of events in one inertial reference frame in terms of those in another. We first examine how position and time coordinates transform between inertial frames according to the view in Newtonian physics.

Then we examine how this has to be changed to agree with the postulates of relativity. Finally, we examine the resulting Lorentz transformation equations and some of their consequences in terms of four-dimensional space-time diagrams, to support the view that the consequences of special relativity result from the properties of time and space itself, rather than electromagnetism. An event is specified by its location and time x , y , z , t relative to one particular inertial frame of reference S.

As an example, x , y , z , t could denote the position of a particle at time t , and we could be looking at these positions for many different times to follow the motion of the particle. For simplicity, assume this relative velocity is along the x -axis. The relation between the time and coordinates in the two frames of reference is then. That is,. These four equations are known collectively as the Galilean transformation. We can obtain the Galilean velocity and acceleration transformation equations by differentiating these equations with respect to time.

We use u for the velocity of a particle throughout this chapter to distinguish it from v , the relative velocity of two reference frames. Differentiation yields. We denote the velocity of the particle by u rather than v to avoid confusion with the velocity v of one frame of reference with respect to the other. Velocities in each frame differ by the velocity that one frame has as seen from the other frame. Observers in both frames of reference measure the same value of the acceleration.

The laws of mechanics are consistent with the first postulate of relativity. Expressing these relations in Cartesian coordinates gives. The left-hand sides of the two expressions can be set equal because both are zero. This follows because we have already shown the postulates of relativity to imply length contraction. Thus the position of the event in S is. The postulates of relativity imply that the equation relating distance and time of the spherical wave front:.

The equations relating the time and position of the events as seen in S are then. This set of equations, relating the position and time in the two inertial frames, is known as the Lorentz transformation. They are named in honor of H. Lorentz — , who first proposed them. Interestingly, he justified the transformation on what was eventually discovered to be a fallacious hypothesis.

Simply interchanging the primed and unprimed variables and substituting gives:. Relativistic phenomena can be analyzed in terms of events in a four-dimensional space-time. When phenomena such as the twin paradox, time dilation, length contraction, and the dependence of simultaneity on relative motion are viewed in this way, they are seen to be characteristic of the nature of space and time, rather than specific aspects of electromagnetism.

In three-dimensional space, positions are specified by three coordinates on a set of Cartesian axes, and the displacement of one point from another is given by:. Something similar happens with the Lorentz transformation in space-time.

The path through space-time is called the world line of the particle. The world line of a particle that remains at rest at the same location is a straight line that is parallel to the time axis. If the particle accelerates, its world line is curved. The increment of s along the world line of the particle is given in differential form as.

This follows from the postulates of relativity, and can be seen also by substitution of the previous Lorentz transformation equations into the expression for the space-time interval:. In addition, the Lorentz transformation changes the coordinates of an event in time and space similarly to how a three-dimensional rotation changes old coordinates into new coordinates:.

Lorentz transformations can be regarded as generalizations of spatial rotations to space-time. It is the same interval of proper time discussed earlier. All observers in all inertial frames agree on the proper time intervals between the same two events.

We can deal with the difficulty of visualizing and sketching graphs in four dimensions by imagining the three spatial coordinates to be represented collectively by a horizontal axis, and the vertical axis to be the ct- axis. Starting with a particular event in space-time as the origin of the space-time graph shown, the world line of a particle that remains at rest at the initial location of the event at the origin then is the time axis.

Any plane through the time axis parallel to the spatial axes contains all the events that are simultaneous with each other and with the intersection of the plane and the time axis, as seen in the rest frame of the event at the origin.

It is useful to picture a light cone on the graph, formed by the world lines of all light beams passing through the origin event A , as shown in Figure 5. Because the event A is arbitrary, every point in the space-time diagram has a light cone associated with it. Consider now the world line of a particle through space-time. Any world line outside of the cone, such as one passing from A through C , would involve speeds greater than c , and would therefore not be possible.

Events such as C that lie outside the light cone are said to have a space-like separation from event A. They are characterized in one dimension by:. An event like B that lies in the upper cone is reachable without exceeding the speed of light in vacuum, and is characterized in one dimension by. The event is said to have a time-like separation from A. Time-like events that fall into the upper half of the light cone occur at greater values of t than the time of the event A at the vertex and are in the future relative to A.

Events that have time-like separation from A and fall in the lower half of the light cone are in the past, and can affect the event at the origin. For any event that has a space-like separation from the event at the origin, it is possible to choose a time axis that will make the two events occur at the same time, so that the two events are simultaneous in some frame of reference.

Therefore, which of the events with space-like separation comes before the other in time also depends on the frame of reference of the observer. Since space-like separations can be traversed only by exceeding the speed of light; this violation of which event can cause the other provides another argument for why particles cannot travel faster than the speed of light, as well as potential material for science fiction about time travel.

Similarly for any event with time-like separation from the event at the origin, a frame of reference can be found that will make the events occur at the same location. Because the relations. All observers in different inertial frames of reference agree on whether two events have a time-like or space-like separation.

The twin paradox discussed earlier involves an astronaut twin traveling at near light speed to a distant star system, and returning to Earth. Because of time dilation, the space twin is predicted to age much less than the earthbound twin. This seems paradoxical because we might have expected at first glance for the relative motion to be symmetrical and naively thought it possible to also argue that the earthbound twin should age less.

To analyze this in terms of a space-time diagram, assume that the origin of the axes used is fixed in Earth. The world line of the earthbound twin is then along the time axis. The world line of the astronaut twin, who travels to the distant star and then returns, must deviate from a straight line path in order to allow a return trip.

As seen in Figure 5. Their paths in space-time are of manifestly different length. This is considerably shorter than the proper time for the earthbound twin by the ratio.

The twin paradox is therefore seen to be no paradox at all. The situation of the two twins is not symmetrical in the space-time diagram. This differs from a rotation in the usual three-dimension sense, insofar as the two space-time axes rotate toward each other symmetrically in a scissors-like way, as shown.

The rotation of the time and space axes are both through the same angle. The mesh of dashed lines parallel to the two axes show how coordinates of an event would be read along the primed axes.

The length scale of both axes are changed by:. As a specific example, consider the near-light-speed train in which flash lamps at the two ends of the car have flashed simultaneously in the frame of reference of an observer on the ground. The space-time graph is shown Figure 5. The world line of both pulses travel along the edge of the light cone to arrive at the observer on the ground simultaneously.

Their arrival is the event at the origin. They therefore had to be emitted simultaneously in the unprimed frame, as represented by the point labeled as t both. In terms of the space-time diagram, the two observers are merely using different time axes for the same events because they are in different inertial frames, and the conclusions of both observers are equally valid. As the analysis in terms of the space-time diagrams further suggests, the property of how simultaneity of events depends on the frame of reference results from the properties of space and time itself, rather than from anything specifically about electromagnetism.

As an Amazon Associate we earn from qualifying purchases. Want to cite, share, or modify this book? This book is Creative Commons Attribution License 4. Skip to Content. University Physics Volume 3 5. Table of contents. Chapter Review. Modern Physics. Answer Key. Describe the Galilean transformation of classical mechanics, relating the position, time, velocities, and accelerations measured in different inertial frames Derive the corresponding Lorentz transformation equations, which, in contrast to the Galilean transformation, are consistent with special relativity Explain the Lorentz transformation and many of the features of relativity in terms of four-dimensional space-time.

Figure 5. The Lorentz transformation equations relate events in the two systems. Use the Lorentz transformation to find the time interval of the signal measured by the communications officer of spaceship S. Express the answer as an equation. The spaceship crew measures the simultaneous location of the ends of the sticks in their frame. Lorentz Transformation and Simultaneity The observer shown in Figure 5.

## Time dilation

In physics and relativity , time dilation is the difference in the elapsed time as measured by two clocks. It is either due to a relative velocity between them "kinetic" time dilation, from special relativity or to a difference in gravitational potential between their locations gravitational time dilation , from general relativity. When unspecified, "time dilation" usually refers to the effect due to velocity. After compensating for varying signal delays due to the changing distance between an observer and a moving clock i. Doppler effect , the observer will measure the moving clock as ticking slower than a clock that is at rest in the observer's own reference frame. In addition, a clock that is close to a massive body and which therefore is at lower gravitational potential will record less elapsed time than a clock situated further from the said massive body and which is at a higher gravitational potential. These predictions of the theory of relativity have been repeatedly confirmed by experiment, and they are of practical concern, for instance in the operation of satellite navigation systems such as GPS and Galileo.

People might describe distances differently, but at relativistic speeds, the distances really are different. Have you ever driven on a road that seems like it goes on forever? If you look ahead, you might say you have about 10 km left to go. If you both measured the road, however, you would agree. Traveling at everyday speeds, the distance you both measure would be the same. You will read in this section, however, that this is not true at relativistic speeds.

## An Example of Time Dilation

In special relativity, an observer in inertial i. A second inertial observer, who is in relative motion with respect to the first, however, will disagree with the first observer regarding which events are simultaneous with that given event. Neither observer is wrong in this determination; rather, their disagreement merely reflects the fact that simultaneity is an observer-dependent notion in special relativity. A notion of simultaneity is required in order to make a comparison of the rates of clocks carried by the two observers.

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*It turns out that as an object moves with relativistic speeds a "strange" thing seems to happen to its time as observed by "us" the stationary observer observer in an inertial reference frame.*

Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. It only takes a minute to sign up. Let there be two bodies a and b. Then let the time interval which has passed on the earth be twice that of the time interval of which passed in space for b, i.

*Special relativity theory introduced an interesting notion about time. Time does not pass at the same rate for moving frames of reference.*

Беккер посмотрел на нее в полном недоумении. - Зачем вам деньги? - спросил. Я не собираюсь оплачивать твое пристрастие к наркотикам, если речь идет об .

*Для этого нужен был политический иммунитет - или, как в случае Стратмора, политическая индифферентность. Сьюзан поднялась на верхнюю ступеньку лестницы.*

Наверное, родители отправили ее сюда по какой-то школьной образовательной программе, снабдив кредитной карточкой Виза, а все кончилось тем, что она посреди ночи вкалывает себе в туалете наркотик. - Вы себя хорошо чувствуете? - спросил он, пятясь к двери. - Нормально, - высокомерно бросила. - А тебе здесь делать нечего.

* Аегорortо. Per favore.*

Martha measures the proper time, in this case. Thus, we can use the time dilation Physics Chapter 26 (Special Relativity) Solutions to Sample Problems.

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Note that time dilation and length contraction are just special cases: it is time-dilation if ∆x = 0 and length contraction if ∆t = 0. 3. The spacetime interval (∆s)2.

Given here are solutions to 24 problems in Special Relativity. The solutions Decay of a π Meson: Time Dilation (“Moving Clocks Run Slow”). Problem