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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

as a free falling object increases speed what is happening to the acceleration
Success Reply
of course g is constant
Alwielland
acceleration also inc
Usman
photo electrons doesn't emmit when electrons are free to move on surface of metal why?
Rafi Reply
What would be the minimum work function of a metal have to be for visible light(400-700)nm to ejected photoelectrons?
Mohammed Reply
give any fix value to wave length
Rafi
40 cm into change mm
Arhaan Reply
40cm=40.0×10^-2m =400.0×10^-3m =400mm. that cap(^) I have used above is to the power.
Prema
i.e. 10to the power -2 in the first line and 10 to the power -3 in the the second line.
Prema
there is mistake in my first msg correction is 40cm=40.0×10^-2m =400.0×10^-3m =400mm. sorry for the mistake friends.
Prema
40cm=40.0×10^-2m =400.0×10^-3m =400mm.
Prema
this msg is out of mistake. sorry friends​.
Prema
what is physics?
sisay Reply
why we have physics
Anil Reply
because is the study of mater and natural world
John
because physics is nature. it explains the laws of nature. some laws already discovered. some laws yet to be discovered.
Yoblaze
is this a physics forum
Physics Reply
explain l-s coupling
Depk Reply
how can we say dirac equation is also called a relativistic equation in one word
preeti Reply
what is the electronic configration of Al
usman Reply
what's the signeficance of dirac equetion.?
Sibghat Reply
what is the effect of heat on refractive index
Nepal Reply
As refractive index depend on other factors also but if we supply heat on any system or media its refractive index decrease. i.e. it is inversely proportional to the heat.
ganesh
you are correct
Priyojit
law of multiple
Wahid
if we heated the ice then the refractive index be change from natural water
Nepal
can someone explain normalization condition
Priyojit Reply
please tell
Swati
yes
Chemist
1 millimeter is How many metres
Darling Reply
1millimeter =0.001metre
Gitanjali
The photoelectric effect is the emission of electrons when light shines on a material. 
Chris Reply
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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