Time dilation modification
Tilting Einstein’s Clock
Rohit
Dhawan
(Ph.D.
scholar)
Department
of Mechanical Engineering, D.C.R.U.S.T Murthal, Sonepat, Haryana, India
Email:-
rohitdhawan28@gmail.com
Abstract:
- This work is just a simple extension to Einstein’s Time dilation
theory. But the outcomes of the work are quite interesting and tells a bit more
about the way, the Time behaves with the addition of motion. Till now we are only
aware of the fact that the time is not absolute and is different for
different observers according to their reference frameworks. The current work
focuses on the rate at which time flows in different directions. The
findings of the work reveal that, in the same reference framework, time
varies at a different flow rate in different directions.
Keywords: -
Time dilation, theory of relativity, time flow rate.
I.INTRODUCTION: - Time theory dates
back to the revolutionary theory put forth by Sir Albert Einstein in 1905
widely known as the “Theory of Relativity”. Let us traverse through this theory
of time. The foundations for the time dilation were
established by the renowned work of Einstein’s special relativity. The theory
follows the concepts of constant speed of light and the other one is that
observers moving at the same speed follow the similar laws of physics. The
time dilation theory uses the thought experiments of a light beam traversing
along a vertical direction in a horizontally moving object. To understand the concept of time dilation
assume a light clock with a light source (at the bottom) and the reflector (at
the top) as shown in Figure.1. This light clock along with observer A is
placed on a moving object such as an airplane or train etc. A similar light
clock is also available in the stationary reference frame of the observer B. The light is emitted from source L and
it reaches the reflector R and bounces back to source L. This one cycle of
light beam corresponds to one tick of the light clock. Suppose the distance
between L and R is ‘D’. Both of the
clocks are well synchronized. In a situation when the moving clock is in the rest
position, then both of the clocks will tick simultaneously. But as soon as the
train starts to move, the situation starts to change and some different
phenomenon comes in to existence. The phenomenon is going
to
be named as time dilation after a few lines. Now for observer A of the moving
train, clock 1 (moving clock) will tick at a normal rate. For the observer, as light is travelling the distance 2*D in one tick of the clock. As we know that
speed = distance/time so time is equal to distance/speed., Time in moving
train = 2D / C where C – speed of light.
Or T = 2D / C - (1)
But
for the stationary observer B, the scenario is somewhat different. What he
observes regarding the moving clock is a different issue. According to observer
B, the light beam of the moving clock is not traversing a distance of 2 D but a
somewhat greater distance as shown in Figure. 2
Now
as the speed of light is constant for both the observers and for observer B
the light beam of moving clock is traveling a greater distance so time for B =
New_distance/speed .
or
T ’ = 2 L / C -
(2)
As
New_distance > distance 2 D so New_time > time. Thus for observer B the
moving clock is showing more time than the clock in his stationary reference the framework, which ultimately means that according to observer B time is passing slowly on moving the train. Suppose
the train is moving with a velocity V, then (for stationary observer B) the
distance travelled from L1 to L2 = {½ * (V*T ’)},
, Distance from L2 to R2 = D, Thus according to Pythagoras theorem
L = {[½ (V*T ’)] 2 + D2}1/2 - (3)
, Distance from L2 to R2 = D, Thus according to Pythagoras theorem
L = {[½ (V*T ’)] 2 + D2}1/2 - (3)
After
eliminating L and D from equation 1, equation 2 and equation 3 the result is
T’ = T / [1 – (V2/ C2)]1/2 - (4)
II. TILTED
LIGHT CLOCK: - Thus from equation (4) it is clear that the time period for the moving
clock is greater than the time period for the stationary clock. This is the main
essence of the theory of relativity which depicts that the time in not absolute
and it changes according to the observer’s frame of reference. Till now it is all
about the work for which the train is moving horizontally and the light beam of the light clock
is moving vertically. Now suppose that Einstein’s clock is not positioned
vertically but is tilted at an angle α with the horizontal direction as shown
in Figure. 3.
Figure
3 Tilted light clock
In
the case train is stationary, the distance between the light source and reflector =
T * C, Thus the vertical distance between the light source and reflector is T*C*Sinα.
Now suppose that the train is moving with any velocity V, in that case, the
distance between light source and reflector is T*C but for the stationary observer
B, the distance traveled by the light beam is T’*C as is shown in the Figure. 4.
Now the horizontal distance traveled is equal to T’ * V and vertical distance
is equal to T * C
* sin α. Thus according to Pythagoras theorem
T ’2
= [(T 2* C2 * sin2 α) + (V2 * T ’2)]
/ C2
|
(5)
|
(T 2* C2
* sin2 α) = T’ 2 –
{(V2 *T ’2) / C2}
|
(6)
|
= T’ 2 (1
– V2 / C2)
|
(7)
|
T’ 2 = (T
2 * Sin2 α) / (1 – V2 / C2
)
|
(8)
|
T’ =
(T *Sin α) / (1 – V2 / C2)1/2
|
(9)
|
T’ =
γ * (T *Sin α)
|
(10)
|
Thus
whatever may be the speed of the train, time in moving train varies according to ‘α’.
Thus time becomes zero for α = 00 and α = 1800 i.e there
is no existence of time in horizontal directions for this case (when the train is
moving horizontally and the clock is also placed horizontally). Similarly, time dilation
will be maximum for α = 900 and it will start decreasing from α = 900
to α = 1800.. Thus it can be concluded that on a moving object the rate
of time flow is different for different directions. Hence we can say that the rate
of time flow is also not constant and it also varies according to the direction or
in other words time flows at different rates in different directions.
III.
Conclusions:-
1. For
calculating the time dilation the formula should be amended to T = γ * (T *Sin
α) instead of T’ = γ * T.
2. Time dilation will be maximum for α
= 900
3.
In the same reference frame, Time flows at different rates in different directions
IV.
References:-
2. https://en.wikipedia.org/wiki/Time_dilation
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