Time Dilation.

Reinforcement:

Hey! guys, from the last blog”!Time.!“, we came to know that Time is just a stubborn illusion. Also how time depends upon the curvature of space-time.

I should first discuss ‘TIME DILATION’ followed by ‘Length Contraction’ because length contraction uses the results of proper and improper time(which we will study later on). Time dilation also requires a good math along with great imagination. So I thought to reinforce your concepts by introducing ‘Michelson-Morely experiment and Gedanken Experiments along with Proper & Improper Time which in fact leads us to time dilation.’

ξThe Michelson-Morley Experiment.

Michelson-Morley Experiment was a scientific attempt to prove the existence of aether: During the 18th-century scientist believed that there exists an invisible substance filled in the empty space. This empty substance was believed to be the medium through which light propagates. 

To prove this, Albert A. Michelson and Edward Morley in 1887 conducted an experiment which included: A light source, 2 mirrors, a detector and a semi-silvered mirror. This was arranged in the following manner:

400px-Michelson-Morley_experiment_(en).svg
Fig.1

It functioned in the following manner,

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Fig.2

Now after forming this setup, they formulated that if an empty space is filled with a substance called as aether, then our earth which is revolving the sun nearly at 100,000 Km/h should feel a drag(the same drag a car feels when it moves on the expressway). So this drag should create an interference at the detector(to know more, why this drag creates an interference see: On The Michelson-Morley Experiment by Khan Academy)

So, when these scientists performed the experiments, they got a fascinating result. The experiment disproved the existence of the aether, as they didn’t found any interference at the detector.

This experiment proved that light does not require medium to travel OR “light travel from nothing (Space).”

 

ξGedanken Experiments.

‘Gedanken’ is a German word for ‘Thought’. So Gedanken Experiment just means a THOUGHT EXPERIMENT.

We use this experiments because of our unableness to perform on practical bases. But this does not mean daydreaming. You can actually prove this type of experiment Mathematically.

♣The need for Gedanken Experiment:

Moving towards Proper and improper time, we first need to create a Gedanken Experiment. So close your eyes(Please don’t literally close your eyes as you have to read further!) and imagine Michelson-Morley Experiment with only one mirror as shown below. Additional to this image a clock to measure the time travelled by the light source from A to B and to A.

 

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Fig.3

Here the time measured by the clock is to the order of -9 nanoseconds. So we take L to be the length of our apparatus which is far too long to give a reading of clocks in minutes(here comes our thought experiment).

Time Dilation- Proper and Improper Time.

–Using our previous Gedanken Experiment

So now you are reinforced with some basic concepts which will help you to understand TIME DILATION.

Let us start with Proper time. In Fig.3 let us suppose the light source sends a signal which illuminates the clock for a while. At this moment our observer notes the status of time. Now this reflected light illuminates the clock again and reaches point A, completing its journey. Here again, our observer notes the time elapsed and calculates the time to be Δt′(‘Δ’ represents elapsed time).

To be more precise,

Δt′=2L­/c [ L=AB and c=speed of light]————-(1)

∴Δt′=2AB/c

This is the case of Proper time where the same clock receives both the signal. Or we can say that the coordination(the cartesian system coordinates)  of the clock remains the same.

2
Fig.4. (a)clock reading when the light leaves ‘A’. (b) clock reading when the light comes back after reflection.

Now let us keep the apparatus of Fig.3 in a moving train and our observer on the ground. Will the same effect happen? Commonly people say yes, because of the light is still following the same path A-B-A. But this is not true is we/observer is on the ground while the train is moving. To our observer on the ground, the light takes an oblique path, which indeed is more than AB as shown in Fig.5

32.png
Fig.5 In this image the apparatus moves from A to A` along with the train with velocity ‘v’. But to the observer, the light follows the oblique path.

So in order to measure the time taken the observer will note the time on the 2 watches fixed at the platform at the starting and ending(you can also imagine at A and A`. But should be fixed on the ground). At this time the observer observes a different time -Fig.6, as the distance increases but the speed of light, remains the same but to an observer, the distance increases. This is the case of Improper Time. Here the coordinates of the clock changes. Hey!, wait for a minute. In this scenario to an observer in the train, the time measured by him is PROPER! So when the observer on the ground sees the clock of one in the train, the one measuring proper time will account less time, in a nutshell, the clock on the moving train runs slow.

2
Fig.6 This image depicts the time observed by the one on the ground.

To be precise,

Δt=AB+BA`/c

But (Bc)²= (AB)²-(AC)²= (BA`)²-(CA`)²

∴Δt′/Δt= 2BC/AB+BA`= 2{(AB)²-(AC)²}^½ /2AB

= √1-(AC/AB)²=  √1-v²/c²———–(2)

Here AC is the horizontal distance travelled at a speed v during the time the light travelled with speed c along the hypotenuse.

Hence we conclude that time does run slowly for a moving object with respect to one who is on the ground at rest. So the Time Dilation is a relative effect.

You can also say that the clock of an observer on the ground at rest runs fast with respect to one in the train. So next time when you are late at work or class, just say, “No I’m on time. It’s just the effect of Time Dilation….!!”

That’s all for this blog. I hope you have it. Write me on Instagram or G-mail for any doubts or edition.

Will write on The ‘LENGTH CONTRACTION’ and ‘LORENTZ TRANSFORMATION’ in my next blog! Like, comment, share and please do Follow for latest blog updates.!!

Thank you!!

Instagram ID a.singapuri__SH

-AS

 

 

 

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