Physics
Length Contraction
Length contraction is a phenomenon in special relativity where an object's length appears shorter when it is moving at a significant fraction of the speed of light relative to an observer. This effect is a consequence of the time dilation and space contraction that occur at high velocities, as described by Einstein's theory of special relativity.
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10 Key excerpts on "Length Contraction"
- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2023(Publication Date)
- Wiley(Publisher)
The relative motion causes a Length Contraction, and L is called a contracted length. A greater speed v results in a greater contraction. The length L 0 of an object measured in the rest frame of the object is its proper length or rest length. Measurements of the length from any reference frame that is in relative motion parallel to that length are always less than the proper length. Be careful: Length Contraction occurs only along the direction of relative motion. Also, the length that is measured does not have to be that of an object like a rod or a circle. Instead, it can be the length (or distance) between two objects in the same rest frame—for example, the Sun and a nearby star (which are, at least approximately, at rest relative to each other). Does a moving object really shrink? Reality is based on observations and mea- surements; if the results are always consistent and if no error can be determined, then what is observed and measured is real. In that sense, the object really does shrink. 1134 CHAPTER 37 Relativity However, a more precise statement is that the object is really mea- sured to shrink—motion affects that measurement and thus reality. When you measure a contracted length for, say, a rod, what does an observer moving with the rod say of your measurement? To that observer, you did not locate the two ends of the rod simul- taneously. (Recall that observers in motion relative to each other do not agree about simultaneity.) To the observer, you first located the rod’s front end and then, slightly later, its rear end, and that is why you measured a length that is less than the proper length. Proof of Eq. 37.2.1 Length Contraction is a direct consequence of time dilation. Con- sider once more our two observers. This time, both Sally, seated on a train moving through a station, and Sam, again on the station platform, want to measure the length of the platform.
eBook - PDF- Stephen Thornton, Andrew Rex, Carol Hood, , Stephen Thornton, Stephen Thornton, Andrew Rex, Carol Hood(Authors)
- 2020(Publication Date)
- Cengage Learning EMEA(Publisher)
Can this really be true? We shall discuss this question again in Section 2.8. Proper length Copyright 2021 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 2.5 Time Dilation and Length Contraction 37 Notice that L 0 . L, so the moving meterstick shrinks according to Frank. This effect is known as length or space contraction and is characteristic of relative motion. This effect is also sometimes called the Lorentz–FitzGerald contrac- tion because Lorentz and FitzGerald independently suggested the contraction as a way to solve the electrodynamics problem. This effect, like time dilation, is also reciprocal. Each observer will say that the other moving stick is shorter. There is no Length Contraction perpendicular to the relative motion, however, because y 9 5 y and z 9 5 z. Observers in both systems can check the length of the other meterstick placed perpendicular to the direction of motion as the metersticks pass each other. They will agree that both metersticks are one meter long. We can perform another gedanken experiment to arrive at the same result. This time we lay the meterstick along the x 9 axis in the moving system K9 (Fig- ure 2.11a). The two systems K and K9 are aligned at t 5 t 9 5 0. A mirror is placed at the end of the meterstick, and a light flash goes off at the origin at t 5 t 9 5 0, sending a light pulse down the x 9 axis, where it is reflected and returned. Mary sees the stick at rest in system K9 and measures the proper length L 0 (which should of course be one meter). Mary uses the same clock fixed at x 9 5 0 for the time measurements.
eBook - PDF- P R Wallace(Author)
- 1991(Publication Date)
- World Scientific(Publisher)
How then do we determine the Length Contraction from the dia-gram? Events simultaneous with P in S' lie along the line PQ. The event at time zero in S', at the position O, is Q. Thus, the length of the rod in S' is the distance QP measured in the direction of the space-axis of S'. This is obviously the same as the distance 0 3 ' , i.e. the length in S' is 3 units. This is the space contraction observed in an object moving with respect to the frame of reference in which the observer is at rest. Note that what we have done here is deduce the space contrac-tion from the time dilation. They are two sides of the same coin. 5.12. A r e Relativistic P h e n o m e n a Real? Do objects moving past us at high speeds really contract in their direction of motion? Do clocks moving very fast with respect to us really slow down? One can reply with a question. What do you mean by really? What I mean is, are relativistic effects some kind Relativity — The Special Theory 117 of conjurer's trick, some sort of illusion, which lead us to see things, or interpret things, which are not objectively happening? If what we observe in nature depends on our frame of reference, does not science become subjective rather than objective? How can we know objective reality? There is no end to these questions. We will pre-empt this game by making an assertion: all rela-tivistic effects are real. Just as real as anything else in physics. Fig. 5.16. If we (O) move past a charge at C, it is moving relative to us and thus produces a magnetic field. Consider the following relativistic effect. If an electric charge is at rest (in a given frame of reference), it produces a static electric field, but no magnetic field. Now take another frame of reference, moving with respect to the charge. To measuring instruments in that frame, the charge is moving, and so produces a magnetic field.
- David Halliday, Jearl Walker, Patrick Keleher, Paul Lasky, John Long, Judith Dawes, Julius Orwa, Ajay Mahato, Peter Huf, Warren Stannard, Amanda Edgar, Liam Lyons, Dipesh Bhattarai(Authors)
- 2020(Publication Date)
- Wiley(Publisher)
Pdf_Folio:899 CHAPTER 37 Relativity 899 FIGURE 37.11 If you mark the front and back at different instances, the distance between the locations is not the penguin’s width. x A (t 0 ) x B (t 1 ) Position at t 1 v Because simultaneity is relative and it enters into length measurements, length should also be a relative quantity. It is. Let L 0 be the length of a rod that you measure when the rod is stationary (meaning you and it are in the same reference frame, the rod’s rest frame). If, instead, there is relative motion at speed v between you and the rod along the length of the rod, then with simultaneous measurements you obtain a length L given by the length‐contraction equation: L = L 0 √ 1 − 2 = L 0 . (37.14) Because the Lorentz factor is always greater than unity if there is relative motion, L is less than L 0 . The relative motion causes a Length Contraction, and L is called a contracted length. A greater speed v results in a greater contraction. The length L 0 of an object measured in the rest frame of the object is its proper length or rest length. Measurements of the length from any reference frame that is in relative motion parallel to that length are always less than the proper length. Be careful: Length Contraction occurs only along the direction of relative motion. Also, the length that is measured does not have to be that of an object like a rod. Instead, it can be the length (or distance) between two objects in the same rest frame — for example, the Sun and a nearby star (which are, at least approximately, at rest relative to each other). Does a moving object really shrink? Reality is based on observations and measurements; if the results are always consistent and if no error can be determined, then what is observed and measured is real. In that sense, the object really does shrink. However, a more precise statement is that the object is really measured to shrink — motion affects that measurement and thus reality.
eBook - PDFFrom Atoms to Galaxies
A Conceptual Physics Approach to Scientific Awareness
- Sadri Hassani(Author)
- 2010(Publication Date)
- CRC Press(Publisher)
L = L 0 r 1 -v 2 c 2 = L 0 γ , (26.2) where γ was introduced in Equation (26.1). L 0 is length measured by an observer stationary relative to the two ends of the length; we call L 0 the rest length . L is length measured by an observer moving relative to the two ends of the length; we call L the moving length . We see that a moving length is shrunk by a factor of p 1 -( v/c ) 2 to an observer with respect to whom the length moves with a speed v . Length Contraction can “explain” why space travel to distant stars is possible, even though light itself may take hundreds of years to travel such distances. The point is that, as a spaceship travels at a speed close to light speed, the distance between Earth and the destination star becomes very short. So, the crew does not see the star 100 light years away, but only 5 light years (if the spaceship is moving fast enough). Example D.26.3 on page 46 of Appendix.pdf provides a sample numerical detail. Example 26.2.1. Having learned about the Length Contraction and having read Example 25.4.2 in Chapter 25, Emmy wants to try to make an ellipse out of a circle by moving it. First she compares lengths differing slightly from 1 meter to see how much a meter stick should shrink before she can actually notice the difference by merely eyeballing it. She decides that she can tell the difference between 100 cm and 95 cm by merely looking at them, i.e., if somebody showed her a meter stick (100 cm long), and a little later a 95-cm stick, she could “remember” that the first stick was longer. Next, she tries to see how fast a circle with a diameter of 100 cm should move horizontally so she could see it as a vertical ellipse. Clearly, it has to move at such a speed that the horizontal diameter shrinks at least to 95 cm. She uses Equation (26.2) with L = 95 cm and L 0 = 100 cm: 95 = 100 p 1 -( v/c ) 2 , whose solution gives v/c = 0 . 312, or v = 0 . 312 c . Plugging in 3 × 10 8 m/s for c , she gets v = 9 .
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Library Press(Publisher)
This phenomenon is called Length Contraction or Lorentz contraction . These effects are not merely appearances; they are explicitly related to our way of measuring time intervals between events which occur at the same place in a given coordinate system (called co-local events). These time intervals will be different in ________________________ WORLD TECHNOLOGIES ________________________ another coordinate system moving with respect to the first, unless the events are also simultaneous. Similarly, these effects also relate to our measured distances between separated but simultaneous events in a given coordinate system of choice. If these events are not co-local, but are separated by distance (space), they will not occur at the same spatial distance from each other when seen from another moving coordinate system. However, the space-time interval will be the same for all observers. The underlying reality remains the same. Only our perspective changes. Causality and prohibition of motion faster than light Diagram 2. Light cone In diagram 2 the interval AB is 'time-like'; i.e. , there is a frame of reference in which events A and B occur at the same location in space, separated only by occurring at different times. If A precedes B in that frame, then A precedes B in all frames. It is hypothetically possible for matter (or information) to travel from A to B, so there can be a causal relationship (with A the cause and B the effect). ________________________ WORLD TECHNOLOGIES ________________________ The interval AC in the diagram is 'space-like'; i.e. , there is a frame of reference in which events A and C occur simultaneously, separated only in space. However there are also frames in which A precedes C (as shown) and frames in which C precedes A. If it were possible for a cause-and-effect relationship to exist between events A and C, then special relativity would be falsified.
eBook - PDFGravity from the Ground Up
An Introductory Guide to Gravity and General Relativity
- Bernard Schutz(Author)
- 2003(Publication Date)
- Cambridge University Press(Publisher)
The Earth has been approaching the experimenter at nearly the speed of light during this time, but even so it cannot have traveled more than 660 m. Yet most of the muons have reached the ground. The inescapable conclusion is that the ground is less than 660 m from the top of the atmosphere, as measured by the ex- perimenter moving with the muons. A length that is more than 20 km as measured by an experimenter at rest on the Earth has contracted to less than 660 m when measured by an experimenter moving at nearly the speed of light. This effect is called the Lorentz–Fitzgerald contraction, because Lorentz and Fitzgerald were the first to propose that it and time dilation actually occurred. The formula is, following the pattern of earlier ones, L moving object = 1 - v 2 /c 2 L object at rest . (16.2) But there is a crucial difference between what Lorentz and Fitzgerald predicted and what Einstein showed really happens. For Lorentz and Fitzgerald, the speed v in this formula was the speed of the object through the ether. Thus, in the Michelson– Morley experiment, they expected that the length of the arm of the interferometer that lay along the direction of motion of the Earth was physically shorter than the other arm, but that this was unfortunately unmeasurable: as soon as Michelson held a ruler up against this arm to measure its length, the rule would contract by the same amount, so the arm would appear to have its rest-length. Nevertheless, to Lorentz and Fitzgerald, the arm “really” was shorter. For Einstein, the length is what the experimenter measures, and the contraction occurs only when the object moves relative to the experimenter who makes the length measurement. This fun- damental difference in interpretation is the main reason that Einstein gets the credit for discovering the contraction effect, even though the mathematical expression is the same as for Lorentz and Fitzgerald, and even though we honor their contribution by naming the effect after them.
eBook - PDF- William Moebs, Samuel J. Ling, Jeff Sanny(Authors)
- 2016(Publication Date)
- Openstax(Publisher)
The distance between the same two events (production and decay of a muon) depends on who measures it and how they are moving relative to it. Chapter 5 | Relativity 199 Proper Length Proper length L 0 is the distance between two points measured by an observer who is at rest relative to both of the points. The earthbound observer measures the proper length L 0 because the points at which the muon is produced and decays are stationary relative to Earth. To the muon, Earth, air, and clouds are moving, so the distance L it sees is not the proper length. Figure 5.9 (a) The earthbound observer sees the muon travel 2.01 km. (b) The same path has length 0.627 km seen from the muon’s frame of reference. The Earth, air, and clouds are moving relative to the muon in its frame, and have smaller lengths along the direction of travel. Length Contraction To relate distances measured by different observers, note that the velocity relative to the earthbound observer in our muon example is given by v = L 0 Δt . The time relative to the earthbound observer is Δt, because the object being timed is moving relative to this observer. The velocity relative to the moving observer is given by v = L Δτ . The moving observer travels with the muon and therefore observes the proper time Δτ. The two velocities are identical; thus, L 0 Δt = L Δτ . We know that Δt = γΔτ. Substituting this equation into the relationship above gives (5.3) L = L 0 γ . Substituting for γ gives an equation relating the distances measured by different observers. Length Contraction Length Contraction is the decrease in the measured length of an object from its proper length when measured in a reference frame that is moving with respect to the object: (5.4) L = L 0 1 − v 2 c 2 where L 0 is the length of the object in its rest frame, and L is the length in the frame moving with velocity v.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
An observer who is in motion with respect to the events and who views them as occurring at different places measures a dilated time interval Δt. The dilated time interval is greater than the proper time interval, according to the time-dilation equation (Equation 28.1). In this expression, υ is the relative speed between the observer who measures Δt 0 and the observer who measures Δt. Δt = Δ t 0 ________ √ ______ 1 − υ 2 __ c 2 (28.1) 28.4 The Relativity of Length: Length Contraction The proper length L 0 between two points is the length measured by an observer who is at rest relative to the points. An observer moving with a relative speed υ parallel to the line between the two points does not measure the proper length. Instead, such an observer meas- ures a contracted length L given by the length-contraction formula (Equation 28.2). Length Contraction occurs only along the direction of the motion. Those dimensions that are perpendicular to the motion are not shortened. The observer who measures the proper length may not be the observer who measures the proper time interval. L = L 0 √ ______ 1 − υ 2 __ c 2 (28.2) 914 CHAPTER 28 Special Relativity 28.5 Relativistic Momentum An object of mass m, moving with speed υ, has a relativistic momentum whose magnitude p is given by Equation 28.3. p = mυ ________ √ ______ 1 − υ 2 __ c 2 (28.3) 28.6 The Equivalence of Mass and Energy Energy and mass are equivalent. The total energy E of an object of mass m, moving at speed υ, is given by Equation 28.4. The rest energy E 0 is the total energy of an object at rest (υ = 0 m/s), as given by Equation 28.5. An object’s total energy is the sum of its rest energy and its kinetic energy KE, or E = E 0 + KE. Therefore, the kinetic energy is given by Equation 28.6.
eBook - PDF- Kenneth S. Krane(Author)
- 2020(Publication Date)
- Wiley(Publisher)
For a derivation of the Lorentz transformation, see R. Resnick and D. Halliday, Basic Concepts in Relativity (New York, Macmillan, 1992). 42 Chapter 2 The Special Theory of Relativity equations of the Lorentz transformation are consistent with the postulates of special relativity. Length Contraction A rod of length L 0 is at rest in the reference frame of observer O ′ . The rod extends along the x ′ axis from x ′ 1 to x ′ 2 ; that is, O ′ measures the proper length L 0 = x ′ 2 − x ′ 1 . Observer O, relative to whom the rod is in motion, measures the ends of the rod to be at coordinates x 1 and x 2 . For O to determine the length of the moving rod, O must make a simultaneous determination of x 1 and x 2 , and then the length is L = x 2 − x 1 . Suppose the first event is O ′ setting off a flash bulb at one end of the rod at x ′ 1 and t ′ 1 , which O observes at x 1 and t 1 , and the second event is O ′ setting off a flash bulb at the other end at x ′ 2 and t ′ 2 , which O observes at x 2 and t 2 . The equations of the Lorentz transformation relate these coordinates, specifically, x ′ 1 = x 1 − ut 1 √ 1 − u 2 ∕c 2 x ′ 2 = x 2 − ut 2 √ 1 − u 2 ∕c 2 (2.24) Subtracting these equations, we obtain x ′ 2 − x ′ 1 = x 2 − x 1 √ 1 − u 2 ∕c 2 − u(t 2 − t 1 ) √ 1 − u 2 ∕c 2 (2.25) O ′ must arrange to set off the flash bulbs so that the flashes appear to be simul- taneous to O. (They will not be simultaneous to O ′ , as we discuss later in this section.) This enables O to make a simultaneous determination of the coordi- nates of the endpoints of the rod. If O observes the flashes to be simultaneous, then t 2 = t 1 , and Eq. 2.25 reduces to x ′ 2 − x ′ 1 = x 2 − x 1 √ 1 − u 2 ∕c 2 (2.26) With x ′ 2 − x ′ 1 = L 0 and x 2 − x 1 = L, this becomes L = L 0 √ 1 − u 2 ∕c 2 (2.27) which is identical with Eq. 2.13, which we derived earlier using Einstein’s postulates.
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