cosmic adventure 5.6 time dilation in relativity

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TIME DILATION IN RELATIVITY

Cosmic Adventure 5.6


Temporal Transform

Not only frame transformation changes the length of an object, but also the timing on the object.


Proper Time

An observer at the rest frame with his single clock made a measurement of the time on another clock. This time interval, 𝑡′′ − 𝑡′ = ∆𝑡0, is called the proper time interval between the events.

𝑡′′𝑡′

Proper time: ∆𝑡0= 𝑡′′ − 𝑡′Rest Frame


Proper Time Equation

To find the relationship between the time separations as measured by Oand O’ , the relativistic way is to subtract two of the Lorentz time transformations:

∆𝑡0= 𝑡′′ − 𝑡′ →𝑡′′ +

𝑣𝑥′′

𝑐2

1 − 𝑣2 𝑐2−

𝑡′ +𝑣𝑥′

𝑐2

1 − 𝑣2 𝑐2

=𝑡′′ +

𝑣𝑥′′

𝑐2− 𝑡′ −

𝑣𝑥′

𝑐2

1 − 𝑣2 𝑐2


Resulting Difference

∆𝑡0=𝑡′′ +

𝑣𝑥′′

𝑐2− 𝑡′ −

𝑣𝑥′

𝑐2

1 − 𝑣2 𝑐2

These time measurements are made in the moving frame. When they are made at the same location, the expression will be reduced to:

∆𝑡 =𝑡′′ − 𝑡′

1 − 𝑣2 𝑐2=

∆𝑡0

1 − 𝑣2 𝑐2= 𝛾∆𝑡0

Where 𝛾 =1

1− 𝑣2 𝑐2


Time Dilation

Since: 1 − 𝑣2 𝑐2 < 1,

∆𝑡 > ∆𝑡0

Being the time interval between the two events measured by O’ is considered to be dilated(enlarged), thereby giving rise to the phenomenon of “time dilation”.


Equation of Time Dilation

∆𝑡 =∆𝑡0

1 − 𝑣2 𝑐2

Observed time Proper time

Lorentz Factor


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

10.0

8.0

6.0

4.0

2.0

1.0

Tim

e di

lati

on

∆𝑡 =∆𝑡0

1 −𝑣2

𝑐2

Velocity as a fraction of the speed of light

𝑣/𝑐


Time Changes due to Frame Transfer

For example, one can place cameras at the location of clock B and at the location of the clock C, and a picture is taken by each camera when the clock C passes clock B.

Each picture will show that the clock C has advanced through ∆𝑡0 while clock B has advanced through ∆𝑡.

∆𝑡 and ∆𝑡0 are related by the time dilation equation.

Clock A Clock BClock B

Clock C

Clock C

𝑣

𝑣Clock C advances through ∆𝑡𝑜

Clock A & B advances through ∆𝑡


Slower Clocks

For the observer at rest on the ground, the object’s clock is running at a lower rate. But for the clock carrier, there is no difference at all. His clock is running at the normal rate as when he is at rest.

No! It is not!

Your clock is slowing down!


Time Lapse by Slower Clock

Once the moving clock re-unites with the ground clock, it will again run at the same rate as the ground clock. It is only when the clock is in motion with respect to the other clock that the phenomenon of time dilation takes place.

Naturally, the slower clock will remain behind by the amount of time it spent.


Time Dilation is Retained by Slower Clock

Relativists believe that if you take two very accurate and well synchronized atomic clocks and put one on a high-speed trip on an airplane. When the plane returned, the clock that took the plane ride was slower.


The Twin Paradox

The wonderful thing is that the returned clock keeps the time it gained in the trip. Anyone who travels with the clock will consequently live longer. This leads to the phenomenon of “Twin Paradox”. It is a paradox because nobody has verified it by trying to live longer this way.


Dr. Einstein will return to earth nearly as young, while his twin on earth will have aged terribly.


Example of Calculation

The earth bound Einstein calculate the time dilation with the equation:

∆𝑡 =∆𝑡0

1 − 𝑣2 𝑐2

where:

∆𝑡 = time observed in the other reference frame

∆𝑡0 = time in observers own frame of reference (rest time)

𝑣 = the speed of the moving rocket

𝑐 = the speed of light in a vacuum


Calculating the Dilation

So in the situation we will let:

𝑣 = .95𝑐

𝑡0 = 10 𝑦𝑒𝑎𝑟𝑠

and we will solve for t which is the time that the earth bound Einstein brother measures.

𝑡 =10

1 −0.95𝑐 2

𝑐2

=10

1 − 0.9025=

10

0.0975

=10

0.3122= 32

𝑇𝑖𝑚𝑒 = 32 𝑦𝑒𝑎𝑟𝑠


Muon Life Time

The extended lifetime of the muon provides another proof of time dilation. But so far the results are not absolutely precise and the relativistic explanation is not too clear. There are other ways of achieving the results and in a more logical way. We will be glad to talk about it in a different session.


Both Answers Real and Correct

Objectively speaking, the observer is seeing two different times.

So which time is correct? According to the relativists, they both are.

The reason is that time is not absolute but is relative, it depends on the relative relationship of the reference frames.

So why then is the phenomenon called a

paradox?


VISONIC TIME DILATION – CLOCKS AT REST

To be continued on：

Cosmic Adventure 5.7

cosmic adventure 5.6 time dilation in relativity

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