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I was trying to understand Lorentz Transformation equation and special theory of relativity, but as I compared the derivation with a thought experiment which I imagined I found the whole Lorentz Transformation Equation fails. The details of the problem is given below. I know I m wrong but I wish you help me to find where I went wrong.

Thought Experiment Details

PLEASE SEE THE FIGURE ATTACHED

Let us consider a spacecraft moving with a velocity v m/s, two observers are there, one inside the spacecraft (observer-1) and other watching spacecraft from earth (observer-2). Two light sources (L1, L2) are emitted inside the spacecraft from point ‘P’ and point ‘A’ simultaneously as shown in figure. After one second (time with respect to observer-2) the spacecraft moves ‘x’ m. Observer-1 measures light source L1 moved a distance ‘a’ and light source L2 a distance ‘b’, but observer-2 measures L1 moved a distance ‘a1’ and L2 a distance ‘b1’.

From ΔABC Lorentz Transformation Equation can be derived;

Ø x = v*t1....................................................................................(1)

Ø a = c*t.......................................................................................(2)

Ø a1 = c*t1...................................................................................(3)

where;

x is the distance travelled by spacecraft in time t1 (t1 is taken as 1 sec for simplicity)

t1 is time w.r.t observer-2

v is velocity of spacecraft

a is distance travelled by L1 in time t as per observer-1

c is speed of light

a1 is the distance travelled by L1 in time t1 as per observer-2

Applying Pythagoras Theorem to ΔABC;

a1^{2}= a^{2}+ x^{2}

(v*t1)^{2}= (c*t)^{2}+ (c*t1)^{2}

─► (c*t)^{2}= (c*t1)^{2}- (v*t1)^{2}

─►t^{2}= t1^{2}(1- v^{2}/c^{2})

t = t1√(1- v^{2}/c^{2})....................................Lorentz Transformation Equation

ConsiderΔPQR;

Since L1 and L2 are emitted at same time the distance travelled by both light will be same after a time interval ‘t’ as measured by observer-1, hence;

a = b

From figure the sides PR = AC

Hence from theΔABC & ΔPQRit is clear that the distance travelled by L1 & L2 measured by observer-2 in time t1 is not equal.

But the distance travelled by L1 & L2 at time ‘t1’ from observer-2 should be equal since speed of light is constant from any frame of reference;

ie;

a1 = c*t1

b1 = c*t1

Since a1≠b1, the only thing which should vary is the speed of light measured from observer-2’s frame of reference.

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# Lorentz Transformation Equation Paradox

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