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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

عند قذف جسم إلى أعلى بسرعة إبتدائية فإنه سيصل إلى ارتفاع معين (أقصى ارتفاع) ثم يعود هابطاً نحو سطح الأرض .   إذا قُذِفَ جسم إلى أعلى ووجد أن سرعته 18 م / ث عندما قطع 1/4 المسافة التي تمثل أقصى ارتفاع سيصله فالمطلوب إيجاد السرعة التي قُذِف بها بالمتر / ث . إن هذه السرعة هي واحدة من الإجابات التالية
Aml Reply
what is light
Ayebanifesunday Reply
light is a kind of radiation That stimulates sight brightness a source of illumination.
Electromagnet radiation creates space 7th, 8th, and 9th dimensions at the rate of c.
That is the reason that the speed of light is constant.
This creation of new space is "Dark Energy".
The first two sets of three dimensions, 1 through 6, are "Dark Matter".
As matter decays into luminous matter, a proton, a neutron, and an electron creat deuterium.
There are three sets of three protons, 9.
There are three sets of three neutrons, 9.
A free neutron decays into a proton, an electron, and a neutrino.
There are three sets of five neutrinoes, 15.
Neutrinoes are two dimensional.
A positron is composed of the first three dimensions.
An electron is composed of the second three dimensions.
What is photoelectric
Hsssan Reply
light energy (photons) through semiconduction of N-P junction into electrical via excitation of silicon purified and cristalized into wafers with partially contaminated silicon to allow this N-P function to operate.
i.e. Solar pannel.
If you lie on a beach looking at the water with your head tipped slightly sideways, your polarized sunglasses do not work very well.Why not?
Rakhi Reply
it has everything to do with the angle the UV sunlight strikes your sunglasses.
this is known as optical physics. it describes how visible light, ultraviolet light and infrared light interact when they come into contact with physical matter. usually the photons or light upon interaction result in either reflection refraction diffraction or interference of the light.
I hope I'm clear if I'm not please tell me to clarify further or rephrase
what is bohrs model for hydrogen atom
Swagatika Reply
what is the value of speed of light
Propessor Reply
1.79×10_¹⁹ km per hour
what r dwarf planet
Sivalakshmi Reply
what is energy
Isiguzo Reply
কাজের একক কী
কাজের একক কী
Energy is ability so capacity to do work.
friction ka direction Kaise pata karte hai
Rahul Reply
friction is always in the opposite of the direction of moving object
A twin paradox in the special theory of relativity arises due to.....? a) asymmetric of time only b) symmetric of time only c) only time
Varia Reply
b) symmetric of time only
fundamental note of a vibrating string
fasoyin Reply
every matter made up of particles and particles are also subdivided which are themselves subdivided and so on ,and the basic and smallest smallest smallest division is energy which vibrates to become particles and thats why particles have wave nature
what are matter waves? Give some examples
mallam Reply
according to de Broglie any matter particles by attaining the higher velocity as compared to light'ill show the wave nature and equation of wave will applicable on it but in practical life people see it is impossible however it is practicaly true and possible while looking at the earth matter at far
a centeral part of theory of quantum mechanics example:just like a beam of light or a water wave
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
at subatomic level wave and matter are associated. this refering to mass energy equivalence
it is key of quantum
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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