Lorentz transformation equations for space and time

Results of Galilean Transformation equations can not be applied for the objects moving with a speed comparative to the speed of the light.

Therefore new transformations equations are derived by Lorentz for these objects and these are known as Lorentz transformation equations for space and time.

Let there are two inertial frames of references S and S’. S is the stationary frame of reference and S’ is the moving frame of reference. At time t=t’=0 that is in the start, they are at the same position that is Observers O and O’ coincides. After that S’ frame starts moving with a uniform velocity v along x axis. Let an event happen at position P in the frame S’. The coordinate of the P will be x’ according to the observer in S’ and it will be x according to O in S.

The frame S’ has moved a distance “vt” in time t (refer figure).

What should be the relation between x and x’? As we can see from the figure that from frame S’

x’ α x – vt

or x’ = k (x – vt)                                                           (1)

where k is constant of proportionality that we will determine.

Similarly from frame S

x = k(x’ + vt’)                                                  (2)

Put equation (1) in (2)

x = k[k(x – vt) + vt’]

or x/k = kx – kvt + vt’

or vt’ = x/k – kx + kvt

or t’ = x/kv – kx + kvt

or t’ = kt – kx (1 – 1/k2)/v                                (3)

Similarly from frame S, time t will be

t = kt’ + kx’ (1 – 1/k2)/v                                               (4)

(This equation can be derived by putting equation 2 in 1 and then solving.)

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Michelson-Morley experiment

Aim of the Michelson-Morley experiment: The Michelson-Morley experiment was done to confirm the presence of hypothetical medium called ether.

Therefore, one question should be there what was ether? Yes, I have used “was ether” not “is ether”. Let us discuss why?

As we have already discussed in earlier articles that there is nothing like absolute rest. Thus the scientists in the 19th century assumed that our universe is filled with hypothetical medium called ether. Ether was supposed to be transparent and highly elastic.

The main objective of this Michelson-Morley experiment was to check the presence of this medium called ether. The aim was supposed to be fulfilled by measuring the velocity of the earth with respect to the ether. If earth is supposed to be propagating through the stationary ether with a uniform velocity and if a beam of light is sent from source to observer towards the direction of the motion of the earth, then it should take more time if sent through the opposite direction. If this time difference can be measured then velocity of earth with respect to ether can be measured.

Experimental arrangement of Michelson-Morley experiment: Continue reading “Michelson-Morley experiment”

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Twin paradox in relativity

Paradox means confusion and meaning of twins you know. This is related with the concept of time dilation in relativity. Suppose there are twins A and B.

In the twin paradox, one of the twins say A was sent to space in a spaceship which is traveling with a speed comparative to the speed of the light. B remains at earth.

So according to the time dilation, time should be dilated (increased) or in other words clock should be moving slower. But out of A and B, whom clock should be slower? In other words, whose age will be different after certain time?

According to B (which remains at earth), age of A will be different as he is travelling with relativistic speed. But according to A, age of B will be different as he is travelling with relativistic speed in opposite direction.

So who is speaking truth? Both are right at their places. This is called twin paradox.

What is your view on this?

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Real life example of time dilation

Real example of time dilation:

As we have already discussed the concept of time dilation. Let us discuss its example:

Decay of µ- mesons:

µ- mesons are the particles formed in the earth atmosphere. The half life time of µ- mesons is 3.1 microseconds.

They travel with the speed of 0.9c.

where c is the speed of the light.

So they must covered the distance d = vt

d = 3.1 x 10-6 x 0.9c = 840m

It means there population should become half after this distance. But this does not happen. Population remains much higher than the half value.

Why this happened? This is because the time here should be dilated time and 3.1 microseconds should be the proper time. This is because the µ- mesons are travelling with speed comparable to the speed of the light.

So t = t’/(1 – v2/c2)

Here t’ = 3.1 microseconds

After solving, we get

t = 7.2 microseconds

thus distance traveled by µ- mesons will be

d = vt

or d = 7.2 x 10-6 x 0.9c

or d = 1920 m

Now when the population is measured after this distance it was approximately half.

It proves that the time dilation is a real effect.

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