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By the end of this section, you will be able to:
  • Explain how time intervals can be measured differently in different reference frames.
  • Describe how to distinguish a proper time interval from a dilated time interval.
  • Describe the significance of the muon experiment.
  • Explain why the twin paradox is not a contradiction.
  • Calculate time dilation given the speed of an object in a given frame.

The analysis of simultaneity shows that Einstein’s postulates imply an important effect: Time intervals have different values when measured in different inertial frames. Suppose, for example, an astronaut measures the time it takes for a pulse of light to travel a distance perpendicular to the direction of his ship’s motion (relative to an earthbound observer), bounce off a mirror, and return ( [link] ). How does the elapsed time that the astronaut measures in the spacecraft compare with the elapsed time that an earthbound observer measures by observing what is happening in the spacecraft?

Examining this question leads to a profound result. The elapsed time for a process depends on which observer is measuring it. In this case, the time measured by the astronaut (within the spaceship where the astronaut is at rest) is smaller than the time measured by the earthbound observer (to whom the astronaut is moving). The time elapsed for the same process is different for the observers, because the distance the light pulse travels in the astronaut’s frame is smaller than in the earthbound frame, as seen in [link] . Light travels at the same speed in each frame, so it takes more time to travel the greater distance in the earthbound frame.

Figure a shows an illustration of an astronaut in the space shuttle observing an analog clock with an elapsed time Delta tau. The details of the clock experiment are also shown as follows: There is a light source, a receiver a short distance to its right, and a mirror centered above them. The vertical distance from the receiver and light source to the mirror is labeled as D. The path of the light from the source, up to the mirror, and back down to the receiver is shown. Figure b shows an observer on earth with an analog clock showing a time interval Delta t. Above the observer are three diagrams showing the clock experiment on the space shuttle at three different times and the path of the light. The light source in the diagram on the left is labeled “beginning event.” The receiver in the diagram on the right is labeled “ending event.” The path of the light forms a straight line going diagonally up and to the right, from the source in the diagram on the left to the mirror in the center diagram, and then another straight line going diagonally down and to the right, from the mirror in the center diagram to the receiver in the diagram on the right. The vertical distance from the receiver to the mirror is labeled D. The horizontal distance from the beginning event to the clock location in the center diagram is labeled L= v Delta t over 2. The horizontal distance from the clock location in the center diagram to the ending event is labeled L. Figure c shows an isosceles triangle with a horizontal base. The triangle is divided by a vertical line from its apex to its base into two identical right triangles with the vertical line forming a side that is shared by the two right triangles. This side is labeled D. The base of the triangle on the left is labeled L= v Delta t over 2. The base of the triangle on the right is labeled L. The hypotenuse of each of the right triangles is labeled s. Above the diagram is the equation s equals the square root of the quantity D squared plus L squared.
(a) An astronaut measures the time Δ τ for light to travel distance 2 D in the astronaut’s frame. (b) A NASA scientist on Earth sees the light follow the longer path 2 s and take a longer time Δ t . (c) These triangles are used to find the relationship between the two distances D and s .

Time dilation

Time dilation is the lengthening of the time interval between two events for an observer in an inertial frame that is moving with respect to the rest frame of the events (in which the events occur at the same location).

To quantitatively compare the time measurements in the two inertial frames, we can relate the distances in [link] to each other, then express each distance in terms of the time of travel (respectively either Δ t or Δ τ ) of the pulse in the corresponding reference frame. The resulting equation can then be solved for Δ t in terms of Δ τ .

The lengths D and L in [link] are the sides of a right triangle with hypotenuse s . From the Pythagorean theorem,

s 2 = D 2 + L 2 .

The lengths 2 s and 2 L are, respectively, the distances that the pulse of light and the spacecraft travel in time Δ t in the earthbound observer’s frame. The length D is the distance that the light pulse travels in time Δ τ in the astronaut’s frame. This gives us three equations:

2 s = c Δ t ; 2 L = v Δ t ; 2 D = c Δ τ .

Note that we used Einstein’s second postulate by taking the speed of light to be c in both inertial frames. We substitute these results into the previous expression from the Pythagorean theorem:

Questions & Answers

Mathematical expression of principle of relativity
Nasir Reply
given that the velocity v of wave depends on the tension f in the spring, it's length 'I' and it's mass 'm'. derive using dimension the equation of the wave
obia Reply
What is the importance of de-broglie's wavelength?
Mukulika Reply
he related wave to matter
Zahid
at subatomic level wave and matter are associated. this refering to mass energy equivalence
Zahid
how those weight effect a stable motion at equilibrium
Nonso Reply
how do I differentiate this equation- A sinwt with respect to t
Evans Reply
just use the chain rule : let u =wt , the dy/dt = dy/du × du/dt : wA × cos(wt)
Jerry
I see my message got garbled , anyway use the chain rule with u= wt , etc...
Jerry
de broglie wave equation
LoNE Reply
vy beautiful equation
chandrasekhar
what is electro statics
fitsum Reply
when you consider systems consisting of fixed charges
Sherly
Diagram of the derive rotational analog equation of v= u+at
Nnamnso Reply
what is carat
Arnulfo Reply
a unit of weight for precious stones and pearls, now equivalent to 200 milligrams.
LoNE
a science that deals with the composition, structure, and properties of substances and with the transformations that they undergo.
LoNE
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Mrs Reply
what chemistry ?
Abakar
where are the mcq
Fred Reply
ok
Giorgi
acids and bases
Navya
How does unpolarized light have electric vector randomly oriented in all directions.
Tanishq Reply
unpolarized light refers to a wave collection which has an equal distribution of electric field orientations for all directions
pro
In a grating, the angle of diffraction for second order maximum is 30°.When light of wavelength 5*10^-10cm is used. Calculate the number of lines per cm of the grating.
Micheal Reply
OK I can solve that for you using Bragg's equation 2dsin0over lander
ossy
state the law of gravity 6
cletus Reply
what is cathodic protection
Ebe Reply
its just a technique used for the protection of a metal from corrosion by making it cathode of an electrochemical cell.
akif
Practice Key Terms 2

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Source:  OpenStax, University physics volume 3. OpenStax CNX. Nov 04, 2016 Download for free at http://cnx.org/content/col12067/1.4
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