# 28.2 Simultaneity and time dilation  (Page 5/8)

 Page 5 / 8

The situation would seem different to the astronaut. Because motion is relative, the spaceship would seem to be stationary and the Earth would appear to move. (This is the sensation you have when flying in a jet.) If the astronaut looks out the window of the spaceship, she will see time slow down on the Earth by a factor of $\gamma =\text{30}\text{.}0$ . To her, the Earth-bound sister will have aged only 2/30 (1/15) of a year, while she aged 2.00 years. The two sisters cannot both be correct.

As with all paradoxes, the premise is faulty and leads to contradictory conclusions. In fact, the astronaut’s motion is significantly different from that of the Earth-bound twin. The astronaut accelerates to a high velocity and then decelerates to view the star system. To return to the Earth, she again accelerates and decelerates. The Earth-bound twin does not experience these accelerations. So the situation is not symmetric, and it is not correct to claim that the astronaut will observe the same effects as her Earth-bound twin. If you use special relativity to examine the twin paradox, you must keep in mind that the theory is expressly based on inertial frames, which by definition are not accelerated or rotating. Einstein developed general relativity to deal with accelerated frames and with gravity, a prime source of acceleration. You can also use general relativity to address the twin paradox and, according to general relativity, the astronaut will age less. Some important conceptual aspects of general relativity are discussed in General Relativity and Quantum Gravity of this course.

In 1971, American physicists Joseph Hafele and Richard Keating verified time dilation at low relative velocities by flying extremely accurate atomic clocks around the Earth on commercial aircraft. They measured elapsed time to an accuracy of a few nanoseconds and compared it with the time measured by clocks left behind. Hafele and Keating’s results were within experimental uncertainties of the predictions of relativity. Both special and general relativity had to be taken into account, since gravity and accelerations were involved as well as relative motion.

1. What is $\gamma$ if $v=0\text{.650}c$ ?

## Solution

$\gamma =\frac{1}{\sqrt{1-\frac{{v}^{2}}{{c}^{2}}}}=\frac{1}{\sqrt{1-\frac{\left(0\text{.}\text{650}c{\right)}^{2}}{{c}^{2}}}}=1\text{.}\text{32}$

2. A particle travels at $1\text{.}\text{90}×{\text{10}}^{8}\phantom{\rule{0.25em}{0ex}}\text{m/s}$ and lives $2\text{.}\text{10}×{\text{10}}^{-8}\phantom{\rule{0.25em}{0ex}}s$ when at rest relative to an observer. How long does the particle live as viewed in the laboratory?

$\Delta t=\frac{{\Delta }_{t}}{\sqrt{1-\frac{{v}^{2}}{{c}^{2}}}}=\frac{2\text{.}\text{10}×{\text{10}}^{-8}\phantom{\rule{0.25em}{0ex}}\text{s}}{\sqrt{1-\frac{\left(1\text{.}\text{90}×{\text{10}}^{8}\phantom{\rule{0.25em}{0ex}}\text{m/s}{\right)}^{2}}{\left(3\text{.}\text{00}×{\text{10}}^{8}\phantom{\rule{0.25em}{0ex}}\text{m/s}{\right)}^{2}}}}=2\text{.}\text{71}×{\text{10}}^{-8}\phantom{\rule{0.25em}{0ex}}\text{s}$

## Section summary

• Two events are defined to be simultaneous if an observer measures them as occurring at the same time. They are not necessarily simultaneous to all observers—simultaneity is not absolute.
• Time dilation is the phenomenon of time passing slower for an observer who is moving relative to another observer.
• Observers moving at a relative velocity $v$ do not measure the same elapsed time for an event. Proper time $\Delta {t}_{0}$ is the time measured by an observer at rest relative to the event being observed. Proper time is related to the time $\Delta t$ measured by an Earth-bound observer by the equation
$\Delta t=\frac{{\Delta t}_{0}}{\sqrt{1-\frac{{v}^{2}}{{c}^{2}}}}={\gamma \Delta t}_{0},$

where

$\gamma =\frac{1}{\sqrt{1-\frac{{v}^{2}}{{c}^{2}}}}.$
• The equation relating proper time and time measured by an Earth-bound observer implies that relative velocity cannot exceed the speed of light.
• The twin paradox asks why a twin traveling at a relativistic speed away and then back towards the Earth ages less than the Earth-bound twin. The premise to the paradox is faulty because the traveling twin is accelerating. Special relativity does not apply to accelerating frames of reference.
• Time dilation is usually negligible at low relative velocities, but it does occur, and it has been verified by experiment.

## Conceptual questions

Does motion affect the rate of a clock as measured by an observer moving with it? Does motion affect how an observer moving relative to a clock measures its rate?

To whom does the elapsed time for a process seem to be longer, an observer moving relative to the process or an observer moving with the process? Which observer measures proper time?

How could you travel far into the future without aging significantly? Could this method also allow you to travel into the past?

## Problems&Exercises

(a) What is $\gamma$ if $v=0\text{.}\text{250}c$ ? (b) If $v=0\text{.}\text{500}c$ ?

(a) 1.0328

(b) 1.15

(a) What is $\gamma$ if $v=0\text{.}\text{100}c$ ? (b) If $v=0\text{.}\text{900}c$ ?

Particles called $\pi$ -mesons are produced by accelerator beams. If these particles travel at $2\text{.}\text{70}×{\text{10}}^{8}\phantom{\rule{0.25em}{0ex}}\text{m/s}$ and live $2\text{.}\text{60}×{\text{10}}^{-8}\phantom{\rule{0.25em}{0ex}}\text{s}$ when at rest relative to an observer, how long do they live as viewed in the laboratory?

$5\text{.}\text{96}×{\text{10}}^{-8}\phantom{\rule{0.25em}{0ex}}\text{s}$

Suppose a particle called a kaon is created by cosmic radiation striking the atmosphere. It moves by you at $0\text{.}\text{980}c$ , and it lives $1\text{.}\text{24}×{\text{10}}^{-8}\phantom{\rule{0.25em}{0ex}}\text{s}$ when at rest relative to an observer. How long does it live as you observe it?

A neutral $\pi$ -meson is a particle that can be created by accelerator beams. If one such particle lives $1\text{.}\text{40}×{\text{10}}^{-\text{16}}\phantom{\rule{0.25em}{0ex}}\text{s}$ as measured in the laboratory, and $0\text{.}\text{840}×{\text{10}}^{-\text{16}}\phantom{\rule{0.25em}{0ex}}\text{s}$ when at rest relative to an observer, what is its velocity relative to the laboratory?

$0.800c$

A neutron lives 900 s when at rest relative to an observer. How fast is the neutron moving relative to an observer who measures its life span to be 2065 s?

If relativistic effects are to be less than 1%, then $\gamma$ must be less than 1.01. At what relative velocity is $\gamma =1\text{.}\text{01}$ ?

$0\text{.}\text{140}c$

If relativistic effects are to be less than 3%, then $\gamma$ must be less than 1.03. At what relative velocity is $\gamma =1\text{.}\text{03}$ ?

(a) At what relative velocity is $\gamma =1\text{.}\text{50}$ ? (b) At what relative velocity is $\gamma =\text{100}$ ?

(a) $0\text{.}\text{745}c$

(b) $0\text{.}\text{99995}c$ (to five digits to show effect)

(a) At what relative velocity is $\gamma =2\text{.}\text{00}$ ? (b) At what relative velocity is $\gamma =\text{10}\text{.}0$ ?

Unreasonable Results

(a) Find the value of $\gamma$ for the following situation. An Earth-bound observer measures 23.9 h to have passed while signals from a high-velocity space probe indicate that $\text{24.0 h}$ have passed on board. (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?

(a) 0.996

(b) $\gamma$ cannot be less than 1.

(c) Assumption that time is longer in moving ship is unreasonable.

definition of mass of conversion
Force equals mass time acceleration. Weight is a force and it can replace force in the equation. The acceleration would be gravity, which is an acceleration. To change from weight to mass divide by gravity (9.8 m/s^2).
Marisa
how many subject is in physics
the write question should be " How many Topics are in O- Level Physics, or other branches of physics.
effiom
how many topic are in physics
Praise
yh I need someone to explain something im tryna solve . I'll send the question if u down for it
a ripple tank experiment a vibrating plane is used to generate wrinkles in the water .if the distance between two successive point is 3.5cm and the wave travel a distance of 31.5cm find the frequency of the vibration
Tamdy
the range of objects and phenomena studied in physics is
what is Linear motion
straight line motion is called linear motion
then what
Amera
linear motion is a motion in a line, be it in a straight line or in a non straight line. It is the rate of change of distance.
Saeedul
Hi
aliyu
Richard
Linear motion is a one-dimensional motion along a straight line, and can therefore be described mathematically using only one spatial dimension
Jason
is a one-dimensional motion along a straight line, and can therefore be described mathematically using only one spatial dimensions.
Praise
what is a classical electrodynamics?
Marga
what is dynamics
Marga
dynamic is the force that stimulates change or progress within the system or process
Oze
what is the formula to calculate wavelength of the incident light
if a spring is is stiffness of 950nm-1 what work will be done in extending the spring by 60mmp
State the forms of energy
machanical
Ridwan
Word : Mechanical wave Definition : The waves, which need a material medium for their propagation, e.g., Sound waves. \n\nOther Definition: The waves, which need a material medium for their propagation, are called mechanical waves. Mechanical waves are also called elastic waves. Sound waves, water waves are examples of mechanical waves.t Definition: wave consisting of periodic motion of matter; e.g. sound wave or water wave as opposed to electromagnetic wave.h
correct
Akinpelu
what is mechanical wave
a wave which require material medium for its propagation
syed
The S.I unit for power is what?
watt
Okoli
Am I correct
Okoli
it can be in kilowatt, megawatt and so
Femi
yes
Femi
correct
Jaheim
kW
Akinpelu
OK that's right
Samuel
SI.unit of power is.watt=j/c.but kw.and Mw are bigger.umots
syed
What is physics
study of matter and its nature
Akinpelu
The word physics comes from a Greek word Physicos which means Nature.The Knowledge of Nature. It is branch of science which deals with the matter and energy and interaction between them.
Uniform
why in circular motion, a tangential acceleration can change the magnitude of the velocity but not its direction
reasonable
Femi
because it is balanced by the inward acceleration otherwise known as centripetal acceleration
MUSTAPHA
What is a wave
Tramsmission of energy through a media
Mateo
is the disturbance that carry materials as propagation from one medium to another
Akinpelu
mistakes thanks
Akinpelu
find the triple product of (A*B).C given that A =i + 4j, B=2i - 3j and C = i + k