1: The spacetime diagram. The spatial axis is horizontal and the time axis vertical; the reference event o is located at the origin. In this frame, events a and b occur in the same place, and events a and c at the same time. Equality of place and simultaneity are frame-dependent properties. 

Contexts in source publication

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... gravity differs considerably from Newton's gravity. Newton described everything with respect to one rigid global frame, in which gravity was an external force from a distance. In contrast, Einstein tells us that locally physics is gravitation-free; i.e., locally, the worldline of a free particle is always straight relative to a free-float frame. But this works only in a limited region of spacetime, and thus we need a variety of local free-float frames. The concept of gravitational force acting on a body is replaced by spacetime geometry; the action of gravity is represented by the object's curvy-yet locally inertial-motion within a curved geometry of spacetime. [77] Spacetime curvature is shown as a change in the separation of two orig- inally parallel worldlines. For instance, consider two balls initially at rest relative to the earth then beginning to fall towards the earth's center. As they fall, their horizontal separation diminishes and their vertical separation widens due to gravitational interactions (as in figure 3.7), and their world- lines are no longer parallel. This tide-producing action results from spacetime curvature in the same way as the curvature of the earth causes the paths of two north-travelling mates to cross at the north pole, even if on the equator they move parallel towards the ...

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... similarity with the cartographer's case extends also to coordinates: in different states of motion-in different frames-two observers disagree about the space and time coordinates of events. Because they also disagree about the squared sum of coordinates, the geometry cannot be Euclidian. To il- lustrate this phenomenon, let us study two spark plug firing events that the laboratory and rocket observer detect. The first plug firing happens when the rocket passes the laboratory-in the reference event o of both frames (see figure 3.4). The laboratory observer finds that the second spark plug fires ten meters from the first firing, just when the rocket passes that point-at event p in figure 3.4 a). Consequently, the rocket observer perceives that the two plug firings happen in the same location ( figure 3.4 b)). Thus the space separation of the events-i.e., their spatial distance-is ten meters for the laboratory observer and zero for the rocket observer. In addition, they do not agree on the time (period) between these ...

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... similarity with the cartographer's case extends also to coordinates: in different states of motion-in different frames-two observers disagree about the space and time coordinates of events. Because they also disagree about the squared sum of coordinates, the geometry cannot be Euclidian. To il- lustrate this phenomenon, let us study two spark plug firing events that the laboratory and rocket observer detect. The first plug firing happens when the rocket passes the laboratory-in the reference event o of both frames (see figure 3.4). The laboratory observer finds that the second spark plug fires ten meters from the first firing, just when the rocket passes that point-at event p in figure 3.4 a). Consequently, the rocket observer perceives that the two plug firings happen in the same location ( figure 3.4 b)). Thus the space separation of the events-i.e., their spatial distance-is ten meters for the laboratory observer and zero for the rocket observer. In addition, they do not agree on the time (period) between these ...

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... similarity with the cartographer's case extends also to coordinates: in different states of motion-in different frames-two observers disagree about the space and time coordinates of events. Because they also disagree about the squared sum of coordinates, the geometry cannot be Euclidian. To il- lustrate this phenomenon, let us study two spark plug firing events that the laboratory and rocket observer detect. The first plug firing happens when the rocket passes the laboratory-in the reference event o of both frames (see figure 3.4). The laboratory observer finds that the second spark plug fires ten meters from the first firing, just when the rocket passes that point-at event p in figure 3.4 a). Consequently, the rocket observer perceives that the two plug firings happen in the same location ( figure 3.4 b)). Thus the space separation of the events-i.e., their spatial distance-is ten meters for the laboratory observer and zero for the rocket observer. In addition, they do not agree on the time (period) between these ...

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... can demonstrate the relativity of simultaneity with the aid of the above equations and a spacetime diagram. The spacetime diagram repre- sents the space and time coordinates x and t of events in a frame, e.g., the laboratory frame. Events in the horizontal line-e.g., a line with t = 0-are simultaneous in that frame. But which events in that diagram are simul- taneous in the rocket frame? The equation for the line of events that are simultaneous with the reference event can be found from equation (3.5) by setting t ? = 0. We get t ? v rel x = 0, and this line is shown in figure 3.9 a). The line also represents the space axis of the rocket frame drawn in the laboratory spacetime diagram. Similarly, the time axis is found by setting x ? = 0 in (3.6). Both axes are shown in figure 3.9 b). The figure shows how the Lorentz transformation turns the axes of a spacetime diagram. Thus, this figure can be considered as a graphical presentation of the Lorentz transfor- mation. Note that both axes of the moving frame tilt towards the lightcone, and that both have an equal tilting angle, a characteristic of all Lorentz transformations. The Lorentz transformation also predicts how velocities are added or, rather, combined. With two frames in relative motion having a relative velocity v rel , the first observer measures the object's velocity to be v ? relative to his frame. The law of the combination of velocities says that the second observer perceives the ...

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... can demonstrate the relativity of simultaneity with the aid of the above equations and a spacetime diagram. The spacetime diagram repre- sents the space and time coordinates x and t of events in a frame, e.g., the laboratory frame. Events in the horizontal line-e.g., a line with t = 0-are simultaneous in that frame. But which events in that diagram are simul- taneous in the rocket frame? The equation for the line of events that are simultaneous with the reference event can be found from equation (3.5) by setting t ? = 0. We get t ? v rel x = 0, and this line is shown in figure 3.9 a). The line also represents the space axis of the rocket frame drawn in the laboratory spacetime diagram. Similarly, the time axis is found by setting x ? = 0 in (3.6). Both axes are shown in figure 3.9 b). The figure shows how the Lorentz transformation turns the axes of a spacetime diagram. Thus, this figure can be considered as a graphical presentation of the Lorentz transfor- mation. Note that both axes of the moving frame tilt towards the lightcone, and that both have an equal tilting angle, a characteristic of all Lorentz transformations. The Lorentz transformation also predicts how velocities are added or, rather, combined. With two frames in relative motion having a relative velocity v rel , the first observer measures the object's velocity to be v ? relative to his frame. The law of the combination of velocities says that the second observer perceives the ...

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... mentioned before, if we consider two fixed events with one placed at the common origin of several spacetime diagrams, in transition from one frame to another the second event draws an invariant hyperbola that opens up along the time axis, as in figure 3.6. The time separation is the smallest in the frame where the events occur in the same place. It is a frame-dependent issue whether the event is located to the left or the right, as seen in figure ...

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... is the invariance of an interval shown in a spacetime diagram? We demonstrate this with an example, where we plot the spacetime diagram of the same events in several frames. At first, we have three observers, i.e., three frames: laboratory, rocket, and super-rocket frames. The super-rocket is one that flies at constant speed in the same direction as the normal rocket but considerably faster. A flash of light is emitted at event o; i.e., at an event when and where the reference clocks of the frames coincide. A portion of the flash is reflected from a mirror traveling with the rocket and returns to the reference clock. In figure 3.5, the path in space of the flash of light is plotted in all three frames. In the rocket frame (figure 3.5 b)) light just goes back and forth to the mirror. The rocket travels so fast that the simple up-down track in its frame appears as a tent profile in the laboratory frame (figure 3.5 a)). In the rocket frame, the emission event o and reception event p take place in the same location, whereas in the laboratory frame they occur far away from each other. It may surprise us that the reception, too, does not occur at the same time in both frames. In fact, if we take for granted that the speed of light (in a vacuum) has the same value for both observers-which, in fact, is one major principle of relativity-this is quite obvious. Because light travels farther when recorded in the laboratory frame than in the rocket frame, the time between events o and p is greater in the ...

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... the context of the spacetime interval, we found that observers in relative motion measure a different frame time between the same events. Figure 3.6 of the flash reflection example showed that the coordinate time is the smallest in a frame where events happen in the same place; this time was called the proper time. Thus each observer always perceives that any other observer measures less time than herself, because the other measures the smallest possible time difference between events near her, i.e., the proper time. This phenomenon whereby an observer finds that another's clock is ticking at a slower rate is called time dilation. A classical example of time dilation, the so-called twin paradox, is given in the next ...

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... frame-dependent entities, as demonstrated in figure 3.12. Spacelike events cannot be causally connected-i.e., they cannot get information from each other-because otherwise cause and effect would be scrambled. Accordingly, no material particle or signal can move between two events connected by a spacelike ...

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... proper time-or cumulative interval-correctly predicts the aging of the traveler traversing the worldline. The shortest path between points in Euclidian space is a straight line, and the other paths are longer. In contrast, in spacetime with Lorentz geometry, the straight worldline between fixed events has the longest proper time and-unexpectedly-the curved worldline is shorter. This is caused by the minus sign in the equation of interval (3.1) (also see figure 3.10 a)). The principle of maximum aging says that, between two specified events, aging attains its maximum uniquely along the straight worldline, and, further, that a free particle follows this worldline of maximal aging. This is valid only in limited regions of spacetime described by special relativity, but as we will see, it can be modified to be sound also in the At event q, the worldline changes its direction ...

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... flash emitted from o expands as a sphere-or a circle in two-dimensional space-in space and traces out a cone opening into the future time direction in spacetime. The future lightcone is shown as an upward opening cone in a three-dimensional spacetime diagram in figure 3.13. Here the third space dimension is suppressed, i.e., two spatial dimensions and one temporal dimension are ...

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... free-fall is not enough to remove all the effects of gravity, be- cause we can find evidence of gravity in its tide-producing action, if we exam- ine a region large enough or for long time enough. Widely separated particles are affected differently by the gravitational interactions of the earth. For ex- ample, two particles released side by side are attracted toward the center of the earth and will, accordingly, move closer together while falling (see figure 3.7 a)). In contrast, vertically separated particles move farther apart as they fall ( figure 3.7 b)). These forces are called tidal, because similar forces from the sun and the moon act on the oceans causing tides. The detection of tidal effects depends on the sensitivity of the measuring instruments. For a given accuracy, a room can always be made so small that these effects cannot be detected in a given time. This room and this period of time are called the free-float reference frame. Definition 3.1. A reference frame is said to be free-float (or inertial or a Lorentz ) reference frame in a certain region of space and time, when every free test particle remains at rest or retains its motion relative to the frame. This test can be done locally, which makes it simple. Tidal forces make special relativity limited, but far from gravitational bodies, the free-float reference frame can constitute a large region of spacetime. Thus special relativity is a local theory, but the localness depends heavily on the location of the region under ...

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... free-fall is not enough to remove all the effects of gravity, be- cause we can find evidence of gravity in its tide-producing action, if we exam- ine a region large enough or for long time enough. Widely separated particles are affected differently by the gravitational interactions of the earth. For ex- ample, two particles released side by side are attracted toward the center of the earth and will, accordingly, move closer together while falling (see figure 3.7 a)). In contrast, vertically separated particles move farther apart as they fall ( figure 3.7 b)). These forces are called tidal, because similar forces from the sun and the moon act on the oceans causing tides. The detection of tidal effects depends on the sensitivity of the measuring instruments. For a given accuracy, a room can always be made so small that these effects cannot be detected in a given time. This room and this period of time are called the free-float reference frame. Definition 3.1. A reference frame is said to be free-float (or inertial or a Lorentz ) reference frame in a certain region of space and time, when every free test particle remains at rest or retains its motion relative to the frame. This test can be done locally, which makes it simple. Tidal forces make special relativity limited, but far from gravitational bodies, the free-float reference frame can constitute a large region of spacetime. Thus special relativity is a local theory, but the localness depends heavily on the location of the region under ...

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... can also divide the worldline into small straight pieces and sum up the spacetime intervals of the pieces. Straight in this case means that velocity along that piece of the worldline remains constant. Segments must be so small that velocity is-within the accuracy-constant between its endpoints, and that the particle acts like a free-float particle along this piece of worldline (see figure 3.10 a)). Because all observers measure the same proper time for individual segments, they also agree on the total proper time. Thus the proper time of a curved worldline has the same value in every inertial reference ...

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... above characteristic of spacetime is often demonstrated with the so- called twin paradox . One twin is sent on a space expedition to a nearby star on a fast spaceship while the other twin remains on the earth. The spaceship's speed is almost that of light, and its journey can be simplified to resemble the kinked worldline in figure 3.10 b). When the traveling twin returns to the earth, he realizes that he has aged less than his stationary brother. Hence, the fast traveling twin ages less. This violates no law but is only a result of Lorentzian geometry, where the traveler's elapsed proper time is his private matter and depends on the spacetime path ...

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... second worldline in figure 3.2 represents a light flash sent from the reference event o. Because light travels one meter in one meter of time, its worldline has been raised 45 degrees from the horizontal axis. As no object can travel faster than light, the inclined angle to the vertical axis of every worldline is less than 45 degrees. A slowly moving object draws an almost vertical worldline, whereas worldlines of faster objects have a more inclined angle to the vertical but yet a slope greater that one, i.e., an angle less than 45 degrees to the vertical axis. [77] If the particle changes its speed (relative to the frame), it draws a world- line that changes its inclination; i.e., the worldline is curved (see figure 3.3). A curved worldline is shown in figure 3.3, where the particle initially slowly time worldline p l i g h t c o n e space o Figure 3.3: Curved worldline of a particle changing its speed. A lightcone limits the worldline's slope at every point. Shown are some acceptable worldlines for the particle after event ...

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... second worldline in figure 3.2 represents a light flash sent from the reference event o. Because light travels one meter in one meter of time, its worldline has been raised 45 degrees from the horizontal axis. As no object can travel faster than light, the inclined angle to the vertical axis of every worldline is less than 45 degrees. A slowly moving object draws an almost vertical worldline, whereas worldlines of faster objects have a more inclined angle to the vertical but yet a slope greater that one, i.e., an angle less than 45 degrees to the vertical axis. [77] If the particle changes its speed (relative to the frame), it draws a world- line that changes its inclination; i.e., the worldline is curved (see figure 3.3). A curved worldline is shown in figure 3.3, where the particle initially slowly time worldline p l i g h t c o n e space o Figure 3.3: Curved worldline of a particle changing its speed. A lightcone limits the worldline's slope at every point. Shown are some acceptable worldlines for the particle after event ...

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... second worldline in figure 3.2 represents a light flash sent from the reference event o. Because light travels one meter in one meter of time, its worldline has been raised 45 degrees from the horizontal axis. As no object can travel faster than light, the inclined angle to the vertical axis of every worldline is less than 45 degrees. A slowly moving object draws an almost vertical worldline, whereas worldlines of faster objects have a more inclined angle to the vertical but yet a slope greater that one, i.e., an angle less than 45 degrees to the vertical axis. [77] If the particle changes its speed (relative to the frame), it draws a world- line that changes its inclination; i.e., the worldline is curved (see figure 3.3). A curved worldline is shown in figure 3.3, where the particle initially slowly time worldline p l i g h t c o n e space o Figure 3.3: Curved worldline of a particle changing its speed. A lightcone limits the worldline's slope at every point. Shown are some acceptable worldlines for the particle after event ...

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... A neutral or unreachable region: no effect whatsoever produced on o can affect what happens in the spacecone of o (e.g., event d in figure 3.13) 4. A material particle emitted inside the past lightcone of o-in the past timecone of o-can affect what is happening in o (e.g., event e in figure 3.13) Future and past timecones together form the timecone of o; i.e., it is composed of the events inside the lightcone. To conclude, Lorentz geometry retains by its nature the cause-and-effect structure of events. Lightcone and the above categories are identical in every overlapping free-float frame; i.e., they are independent of the state of motion of the inertial ...

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... A neutral or unreachable region: no effect whatsoever produced on o can affect what happens in the spacecone of o (e.g., event d in figure 3.13) 4. A material particle emitted inside the past lightcone of o-in the past timecone of o-can affect what is happening in o (e.g., event e in figure 3.13) Future and past timecones together form the timecone of o; i.e., it is composed of the events inside the lightcone. To conclude, Lorentz geometry retains by its nature the cause-and-effect structure of events. Lightcone and the above categories are identical in every overlapping free-float frame; i.e., they are independent of the state of motion of the inertial ...

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... are some established conventions for drawing spacetime diagrams. One event is chosen as the reference event, and thus it is given a zero position in space and zero time. Hence it is located at the origin of the spacetime map. Further, it is customary to choose spatial location as the horizontal axis and time as the vertical axis. Figure 3.1 shows an example of a spacetime map and events placed in the same place or at the same time. Additionally, in spacetime diagrams, it is customary to use the same unit for both spatial distance and time. In fact, henceforth we choose to carry on the same custom and place space and time on an equal footing. We are familiar with the fact that years are used to measure distance, namely light years. A light year is the distance that light travels in empty space in one year. For our purposes, this is an oversized unit, but, fortunately, in the same manner meter can be used as the unit of time. One meter of time is the time that light takes to proceed one meter in a cavity. In conventional units, it is 1/c = 1/(2.998 ? 10 8 ) s ? 3.336 ? 10 ?9 s. The speed of light (in a vacuum) in meters per second, denoted by c, is only a conversion factor between meters and seconds with no deeper physical significance. In fact, c is only a historical accident in humankind's choice of units. Note that the speed of light in meters per meter of time is a unit-free constant and has a unit value. [77] 3.1.3 Worldline A spacetime diagram can give a global picture of all significant events and, especially, if all of them come from the history of one particular particle, we can see the particle's history of travel. The thread connecting the events of the particle's history is the worldline of the particle. A worldline has its own unique existence in spacetime, independent of any frame by which we may choose to describe it. The same applies also to events. In a spacetime diagram, a worldline can be represented by a line, but, strictly speaking, it is not the worldline itself but just an image of the worldline in the spacetime diagram. Figure 3.2 shows two worldlines in a spacetime diagram. The frame where the diagram is plotted is chosen so that two worldlines cross at the event of zero coordinates. The worldline that passes through event a represents a particle traveling at constant speed relative to the frame. The particle is sent from the reference event o and passes through several events shown in the figure. The line drawn through the events illustrates the worldline of the particle. Figure 3.2: Worldlines of a particle (through event a) and light (b). The particle passes through several events, which form a worldline. A light- flash's worldline inclines at the unit slope, i.e., at an angle of 45 degrees to the horizontal ...

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... are some established conventions for drawing spacetime diagrams. One event is chosen as the reference event, and thus it is given a zero position in space and zero time. Hence it is located at the origin of the spacetime map. Further, it is customary to choose spatial location as the horizontal axis and time as the vertical axis. Figure 3.1 shows an example of a spacetime map and events placed in the same place or at the same time. Additionally, in spacetime diagrams, it is customary to use the same unit for both spatial distance and time. In fact, henceforth we choose to carry on the same custom and place space and time on an equal footing. We are familiar with the fact that years are used to measure distance, namely light years. A light year is the distance that light travels in empty space in one year. For our purposes, this is an oversized unit, but, fortunately, in the same manner meter can be used as the unit of time. One meter of time is the time that light takes to proceed one meter in a cavity. In conventional units, it is 1/c = 1/(2.998 ? 10 8 ) s ? 3.336 ? 10 ?9 s. The speed of light (in a vacuum) in meters per second, denoted by c, is only a conversion factor between meters and seconds with no deeper physical significance. In fact, c is only a historical accident in humankind's choice of units. Note that the speed of light in meters per meter of time is a unit-free constant and has a unit value. [77] 3.1.3 Worldline A spacetime diagram can give a global picture of all significant events and, especially, if all of them come from the history of one particular particle, we can see the particle's history of travel. The thread connecting the events of the particle's history is the worldline of the particle. A worldline has its own unique existence in spacetime, independent of any frame by which we may choose to describe it. The same applies also to events. In a spacetime diagram, a worldline can be represented by a line, but, strictly speaking, it is not the worldline itself but just an image of the worldline in the spacetime diagram. Figure 3.2 shows two worldlines in a spacetime diagram. The frame where the diagram is plotted is chosen so that two worldlines cross at the event of zero coordinates. The worldline that passes through event a represents a particle traveling at constant speed relative to the frame. The particle is sent from the reference event o and passes through several events shown in the figure. The line drawn through the events illustrates the worldline of the particle. Figure 3.2: Worldlines of a particle (through event a) and light (b). The particle passes through several events, which form a worldline. A light- flash's worldline inclines at the unit slope, i.e., at an angle of 45 degrees to the horizontal ...

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... are some established conventions for drawing spacetime diagrams. One event is chosen as the reference event, and thus it is given a zero position in space and zero time. Hence it is located at the origin of the spacetime map. Further, it is customary to choose spatial location as the horizontal axis and time as the vertical axis. Figure 3.1 shows an example of a spacetime map and events placed in the same place or at the same time. Additionally, in spacetime diagrams, it is customary to use the same unit for both spatial distance and time. In fact, henceforth we choose to carry on the same custom and place space and time on an equal footing. We are familiar with the fact that years are used to measure distance, namely light years. A light year is the distance that light travels in empty space in one year. For our purposes, this is an oversized unit, but, fortunately, in the same manner meter can be used as the unit of time. One meter of time is the time that light takes to proceed one meter in a cavity. In conventional units, it is 1/c = 1/(2.998 ? 10 8 ) s ? 3.336 ? 10 ?9 s. The speed of light (in a vacuum) in meters per second, denoted by c, is only a conversion factor between meters and seconds with no deeper physical significance. In fact, c is only a historical accident in humankind's choice of units. Note that the speed of light in meters per meter of time is a unit-free constant and has a unit value. [77] 3.1.3 Worldline A spacetime diagram can give a global picture of all significant events and, especially, if all of them come from the history of one particular particle, we can see the particle's history of travel. The thread connecting the events of the particle's history is the worldline of the particle. A worldline has its own unique existence in spacetime, independent of any frame by which we may choose to describe it. The same applies also to events. In a spacetime diagram, a worldline can be represented by a line, but, strictly speaking, it is not the worldline itself but just an image of the worldline in the spacetime diagram. Figure 3.2 shows two worldlines in a spacetime diagram. The frame where the diagram is plotted is chosen so that two worldlines cross at the event of zero coordinates. The worldline that passes through event a represents a particle traveling at constant speed relative to the frame. The particle is sent from the reference event o and passes through several events shown in the figure. The line drawn through the events illustrates the worldline of the particle. Figure 3.2: Worldlines of a particle (through event a) and light (b). The particle passes through several events, which form a worldline. A light- flash's worldline inclines at the unit slope, i.e., at an angle of 45 degrees to the horizontal ...

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... of gravity. The worldline of light has zero proper time. The same worldline is unattainable for any material particle, but the speed of light (in a vacuum) can be approached arbitrarily closely. Let us demonstrate this with the back- and-forth travel shown in figure 3.10 a). As a limiting case when turnaround deacceleration and acceleration happen in an increasingly shorter and shorter period, the worldline turns into a kinked worldline, as in figure 3.10 b). If the traveling speed is sufficiently close to the speed of light, the lapse of proper time along back-and-forth journeys with a kinked worldline can be made as short as we want. This can be noticed from the Lorentz factor, which in- creases without limit when the speed of light is approached. The Lorentz factor expresses the ratio of the laboratory time and traveler's proper ...

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... of gravity. The worldline of light has zero proper time. The same worldline is unattainable for any material particle, but the speed of light (in a vacuum) can be approached arbitrarily closely. Let us demonstrate this with the back- and-forth travel shown in figure 3.10 a). As a limiting case when turnaround deacceleration and acceleration happen in an increasingly shorter and shorter period, the worldline turns into a kinked worldline, as in figure 3.10 b). If the traveling speed is sufficiently close to the speed of light, the lapse of proper time along back-and-forth journeys with a kinked worldline can be made as short as we want. This can be noticed from the Lorentz factor, which in- creases without limit when the speed of light is approached. The Lorentz factor expresses the ratio of the laboratory time and traveler's proper ...

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... cone opening downward in the diagram 3.13 traces the history of an incoming circular pulse perfectly focused to converge on the event o. 2. On the future lightcone of o, a light ray emitted at o can affect-with no time to spare-what will happen (e.g., event b in figure ...

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... 3.1 (Relativity of simultaneity). a) Two events that lie along the direction of relative motion between two frames cannot be simultaneous as measured in both frames. b) Two events with a separation purely transverse Figure 3.8: Einstein's train paradox. Lightning strikes both ends of a rapid train, and two observers see the flashes. The ground observer sees the flashes at the same time and concludes that the lightnings are simul- taneous. The train observer moves towards the flash coming from the front and moves away from the rear flash. Thus she perceives the front flash first and disagrees about the simultaneity of the lightnings with the ground observer. [77] to the direction of relative motion and simultaneous in either frame are si- multaneous in ...

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... twin paradox is not a paradox but a misunderstanding of the relativ- ity of simultaneity. Because the state of motion on the outward and inward trips is different, we must consider them in separate frames. On the outward trip, the earth observer detects that the time progression of the spaceship observer is slower than his, whereas the spaceship traveler observes the op- posite. We end up having a similar outcome also on the inward trip. Now where is the catch when each observer thinks that the other measures less time than he? It is in that the outward and inward observer have a different line of simultaneity, as seen in figure 3.11. There is a vast difference between the earth time of the earth bound events a and b, which are simultaneous with the turnaround event q in the outward and inward frames, respectively. During the U-turn, the place of simultaneity sweeps a large segment of the life line of the stationary twin. That jump in the simultaneity of the traveler explains the supposed paradox. For a full explanation of frame change we need general relativity or a series of frames at different speeds ...

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... to the right but later slows down and comes to rest at event o. Subsequently, the particle accelerates to the left. At every point, the parti- cle's worldline angle to the vertical is limited below 45 degrees. The lines at +45 ? and ?45 ? angles to the vertical axis are called lightcones, because light travels along these lines. In three-and four-dimensional cases, these lines expand into two coaxial cones with their sharp ends, the apexes, facing each other-a shape which is known as the dual cone. Lightcones are shown at events p and o in figure 3.3 with some available worldlines after event ...

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... its worldline is vertical. In contrast, the rocket moves to the right at constant velocity, and thus its worldline tilts an angle relative to vertical. In figure 3.4 b) the same worldlines are drawn relative to the rocket frame and now rocket observer is immovable. For simplicity, let us assume that both frames have the same reference event. The conclusion is that the pointing directions of observers' time axes differ from each other; i.e., they do not agree on the direction of the time axis. Later we find out that they disagree also about the direction of the spatial axis. Situation is somewhat identical to the cartographer's ...

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... us begin by considering two cartographers, who are making maps of (mainly) the same area. They both have the same unit for south-north and west-east directions but they disagree about the direction of north. We ease the situation and suppose that they both have the same physical position corresponding to the zero coordinates. Clearly, they disagree about the coor- dinates of most locations, but fortunately they agree the distances between points; i.e., they agree about the square sum of coordinate separations. This finding is one of the foundations of Euclidian geometry. [77] Similarly, let us study two observers making spacetime maps of their neighborhoods; the first observer stays in the laboratory and the second travels in a rocket with constant velocity relative to the laboratory. In figure 3.4 a) the worldlines of both these observers are represented in the laboratory frame. Because the laboratory observer do not move with respect to the Worldlines of laboratory and rocket observers in a) the labo- ratory frame and in b) the rocket frame. Two spark plug firing events (o and p) are shown in both frames. In the rocket frame the events happens in the same location, but in laboratory frame they have a nonzero space separation. Additionally, the period of time between the events o and p is shorter in the rocket frame compared to the laboratory ...