Proof that interpretation of time dilation is wrong in special theory

of relativity and mass energy equivalence(E=mc2) based on special

theory of relativity is also wrong or proof that some/whole part of

special theory of relativity is wrong

First we will discuss derivation of special theory of relativity in order to understand the faults in the

derivation of special theory of relativity.

Derivation of special theory of relativity and faults/errors in derivation of special theory of

relativity:

Assumption/fundamental postulate of special relativity: ‘Speed of light is same in all reference

frames regardless of the speed of observer and the speed of source of light.’

[Note:

This fundamental assumption/postulate stated just above is actually a wrong assumpti on. It can be

proven that ‘speed of light can be different in different reference frames depending on the speed of

observer and the speed of source of light’ using Maxwell’s equation and some mathematical

analysis. Such a mathematical analysis is given below for your reference.

We will try to derive equation of speed of light emitted by source moving with respect to a particular

reference frame r2. Suppose a co-ordinate system is fixed in reference frame r1 of source in such a

way that the source of electromagnetic radiation is at origin of the co-ordinate system and also the

source and the reference frame r1 in which the source resides are moving with respect to another

reference frame r2 and this motion of reference frame r1 with respect to reference frame r2 is in

positive x-direction with speed v or in other words source is moving with speed v with respect to

reference frame r2 in positive x-direction. Suppose wave propagates in positive x-direction, E-field

oscillations are in y-direction and H-field oscillations are in z-direction and also E-field variations are

sinusoidal with respect to co-ordinate system fixed in reference frame r1.

Therefore for co-ordinate system fixed in reference frame r1:

Assume , and |

| (1)

Using Maxwell’s equation, we will find B.

(2)

Finding from assumed values of Ex, Ey, and Ez, we get

|

→

→

→

| (3)

Therefore from Maxwell’s equation

(4)

Integrating and finding B,

B (0, 0,

) (5)

The phase velocity is the velocity of point on wave that moves with wave at constant phase. In other

words, phase velocity is the velocity with which sine wave peaks are propagated. In our case phas e

velocity is the speed of propagation of electromagnetic wave. To obtain equation of propagation

speed of wave/phase velocity, the phase of the point on wave should be regarded as a constant.

With respect to reference frame r1:

(1.a)

Differentiating it, we get

Propagation speed with respect to reference frame r1

(1.b)

But for reference frame r2 the distance in x direction increases by the amount because of

motion of reference frame r1 with respect to reference frame r2 in positive x-direction with speed v.

Therefore denoting the x-co-ordinate of the co-ordinate system fixed in r2 as and assuming

when , we get

(1.c)

Therefore to get the velocity of wave with respect to reference frame r2: From equation (1.a) and

(1.c), we get

Differentiating it to get phase velocity with respect to reference frame r2, we get

(6)

Now, we have

(7)

For reference frame r1: (Using the co-ordinate system fixed in r1 as described before)

From equations (5) & (7)

(8)

Finding from equation (8), we get

|

→

→

→

|

(9)

From equations (2) & (7)

| | |

|

|

|

| | (10)

From

and equation (7)

| | |

|

|

|

| | (11)

Multiplying equation (10) and (11), we get,

|

|

| ||

|

| | (12)

From equations (4) and (7)

|

|

(13)

Using equations (1), (3), (9), and (13) for putting values into equation (12), we get

(14)

√

(15)

From equation (1.b) and (15),

√

(16)

But speed of light with respect to reference frame r2, using equation (6) and (16):

√ (17)

Hence above mentioned particular example shows that the simple Galilean velocity addition works

with electromagnetic waves and light also. Note that we had taken only a simple case above bu t

generalization of equation (17) can be done and that generalization can be proved also by doing

appropriate similar analysis. In general it can be proved that in vector form for electromagnetic

waves

→

→

→

Above mentioned example and analysis simply proves that velocity addition is wrong in special

theory of relativity. And very large velocities such as the velocity of light can also be added by simple

conventional vector algebra considering Galilean addition of velocities. It also proves that velocity of

light can be different in different reference frames moving with different velocities. Hence the

fundamental postulate of special theory of relativity that ‘speed of light is same in all reference

frames regardless of their velocities’ is proved wrong. Also, the maximum speed limit may not exist.

Hence the sentence ‘Nothing can exceed speed of light’ can also be wrong.]

Apart from this wrong assumption/postulate assumed in special theory of relativity, there is another

error in derivation of special theory of relativity which we shall discuss below.

Einstein did a thought experiment which is given below.

Einstein’s thought experiment:

Suppose a light ray travels up and down in a reference frame r1 as shown in figure just above and

reference frame r1 is travelling with respect to reference frame r2 with speed v in horizontal

direction. In reference frame r1 the light ray originates from point A, reaches point M where it gets

reflected back due to mirror kept there and again terminates at A.

For reference frame r1, suppose total time flow during the ray’s travel from A to M is . Therefore

according to the assumption/postulate (which is actually a wrong assumption) assumed by Einstein

that speed of light is same in all reference frames and it equals c,

(18)

Suppose for reference frame r2, total time flow during the ray’s travel from A to M is In

reference frame r2, the path of the light ray will have a vertical as well as horizontal component as

shown in figure just below due to horizontal motion of reference frame r1 with respect to reference

frame r2. In other words, the light ray will travel some horizontal distance due to horizontal

speed v of reference frame r1 with respect to reference frame r2 and vertical distance h in reference

frame r2 which is shown in figure given just below.

Here also according to the wrong assumption assumed by Einstein, the speed of light should be c.

In time the light ray travels AM distance.

√

(19)

From (18) and (19)

√

(

)

(

)

√

(20)

Equation (20) is the equation of time dilation derived by Einstein. Limits/Drawbacks/Faults of this

equation derived by Einstein and the derivation are discussed below.

Limits/Drawbacks/Errors/Faults of the derivation done by Einstein:

The fundamental postulate that ‘speed of light is same in all reference frames’ is wrong

Apart from the error of assuming the wrong assumption, the derivation of special theory of

relativity performed by Einstein is only one simple case of the general derivation that can be

performed. The method of derivation performed by Einstein is only applicable to the

situation where the direction in which light travels with respect to the reference frame of

source of light (reference frame r1) is perpendicular to the direction in which reference

frame of source (reference frame r1) travels with respect to another reference frame

(reference frame r2). In other words, method of derivation done by Einstein is only

applicable to the situation in which light travels only up-down (vertical) in reference frame r1

when reference frame r1 is travelling in horizontal direction with respect to reference frame

r2. In reality light does not always travel only in vertical direction for which Einstein did

derivation. In reality, light can travel in any direction including the only direction for which

Einstein did derivation. Method of derivation done by Einstein is not applicable to all

directions but it is applicable to only the case in which light travels in perpendicular

direction. The general derivation is possible by specifying the direction with an angle . By

denoting the direction using a variable , we can obtain general result in terms of . And by

putting specific value of

in that general solution, for which Einstein did derivation,

we can obtain equation derived by him. Such a general derivation using the wrong

assumption assumed by Einstein is given below in which light is assumed to travel at an

angle with the horizontal in reference frame r1 (with respect to reference frame r1) and

reference frame r1 is assumed to travel in horizontal direction with respect to reference

frame r2.

Now, we will do another thought experiment (more general than the same done by Einstein)

assuming that ‘light travels at constant speed c in all reference frames’ (which is actually a wrong

assumption but we are using it here to explain further faults/logical errors in derivation of special

theory of relativity apart from the mistake of assuming the wrong assumption. It means that even if

you accept the assumption assumed by Einstein to be true (which is actually false and should not be

accepted by you as true assumption), there are further logical errors in derivation of special theory

of relativity that will force you to reject the theory). Suppose reference frame r1 travels at speed v

with respect to reference frame r2 as shown in figure below, and a light ray travels from point C to A

where a mirror reflects it and sends it to D. Thus light ray travels from C to A to D at an angle with

respect to horizontal in reference frame r1.Suppose time flow during the ray reaches from C to D in

reference frame r1 is Δt1 and the time flow during the ray reaches from C to D in reference frame r2 is

Δt2.

Now,

(21)

(22)

Now, for reference frame r1:

(23)

Now, for reference frame r2:

For reference frame r2, horizontal distance y and light path x increases due to the motion of

reference frame r1 with respect to reference frame r2. From Δ ABC (or Δ ABD) in figure just above,

the increase in the horizontal distance can be found by considering the time interval equal to the

time taken for ray to reach from C to A (or A to D) which is equal to

for reference frame r2.

Therefore increase in the horizontal distance

(24)

And we have

(25)

Applying Pythagorean relation for reference frame r2, we get

(26)

From equations (21),(24) ,(25)and (26), we get

(

)

(

)

(

)

(

)

cot

cot

Now,1+cot2θ=cosec2θ

cosec

(

)

From equation(23) putting

(

)

(

)

(

)

(

)

(27)

The above equation can be considered a quadratic equation of variable (

). Solving for it, using

equation √

√

s in

(A)

This equation shows that for different values of θ and v, there can be different values of the ratio

(

). For any specific reference frame, we know that there can be constant v. But, there can exist

many values of θ possibly infinite because light can travel in any direction in any reference frame.

There can be at least one value of θ for each direction in which light travels.

If you put

rad in equation (A), you will get the equation of time dilation derived by Einstein.

The equation (A) shows that interpretation of time dilation (which was obtained by considering

only

rad) in Einstein’s special theory of relativity was wrong. In wrong interpretation of time

dilation in special theory of relativity, it was derived that

√

which was the case

obtained for

rad only and it disregarded the other infinite possible values of that can exist. It

means that Lorentz transformations are also wrong and wrongly derived.

Based on the wrong interpretation of time dilation done previously in relativity and wrongly

derived/assumed Lorentz transformations, it was wrongly assumed that,

√

From which mass-energy equivalence i.e. E=mc2 equation was derived which should also be wrong

because of the wrong base from which it was derived. The formal proof that E=mc2 is wrong is given

below.

Therefore interpretation of time dilation and energy mass equivalence was wrong in special theory

of relativity.

If we continue analysis which was done previously by Einstein in special theory of relativity, we

should get

√

sin

(28)

which again depends on θ and does not make sense.

From above mentioned equation (28), if θ

radians (which can always happen in infinite number

of ways)

√

(29)

Squaring both sides and rearranging we get,

(30)

Here and c are assumed constants.Doing differentiation, we get

(B)

Now, if we consider work done and energy as same quantity (however they can be treated

differently:energy can be associated with any change and work done can be associated with desired

change.Here we will consider them same quantity)

Work done as per conventional definition=force distance. Treating work done and energy as same

quantity

energy ∫ and

(C)

Putting in equation (C) to convert formula in to formula in , we get

(D)

From equation (B) and (D)

(E)

Integrating both sides

Suppose , then differentiating it , we get which is completely

opposite to equation (E). Therefore . Similarly suppose then

differentiating it, we get which is also completely opposite to equation (E). Hence

. Therefore the concept of energy mass equivalence can/may be wrong. There may be

another way of similar proof using equation(28) instead of equation (29) which may include terms in

θ and sign instead of sign in those equations which may show that energy and mass are

different quantities. Therefore energy and mass should be treated as different quantities and they

are not equivalent. Mass resists change while energy induces change when it is not binding energy.

The interpretation of time dilation based on special theory of relativity was wrong, although time

dilation may or may not exist and it may depend on some other factors such as the amount of

mass(inertial charge), energy(excitation charge) etc.

Conclusions: It just reveals that the way in which special theory of relativity was derived is simply

wrong. First of all the fundamental assumption in special theory of relativity that speed of light is

same in all reference frames regardless of speed of observer and speed of source is completely

wrong. But I have assumed the wrong assumption in this proof just to show that there are serious

logical errors like assuming

rad only in further derivation of special theory of relativity apart

from error of assuming the wrong assumption. Hence the double mistakes committed by Einstein in

deriving special theory of relativity are revealed here. First he assumed the wrong assumption that

speed of light is same in all reference frames and second, even after assuming the wrong

assumption, he committed logical error of assuming

radians only in further derivation from

the wrong assumption. Here note that this proof should be used only for describing the mistakes

committed by Einstein in deriving special theory of relativity and to show that special theory of

relativity is wrong. It should not be used for other purposes. One example of such a purpose i s

making new interpretation of time dilation from the equation (A) which should not be done because

some equations like equation (A) derived here are as wrong as special theory of relativity because

the equations derived here are based on the wrong assumption of the special theory of relativity.

The purpose of this proof is only to show the logical error in derivation of special theory of relativity

apart from the error of assuming wrong assumption. This proof serves its purpose (revealing logical

error in derivation of special theory of relativity) despite it is based on the wrong assumption

contained in special theory of relativity.

Some interpretations based on special theory of relativity are wrong (described above).

Lorentz transformations are wrong and wrongly derived/assumed.

Mass and energy should be treated differently as different physical quantities. Mass energy

equivalence can/may be wrong.

FINALLY YOU CAN SEE THE PROOF BELOW WHICH DERIVES THE TRUE RESULT ASSUMING THE TRUE

ASSUMPTION AND WHICH ALSO VERIFIES THE WRONGNESS OF FUNDAMENTAL POSTULATE OF

SPECIAL THEORY OF RELATIVITY

TRUE DERIVATION BASED ON TRUE ASSUMPTION WHICH CLEARS ALL DOUBTS

Suppose reference frame r1 travels at speed v with respect to reference frame r2 as shown in figure

below, and a light ray travels from point C to A where a mirror reflects it and sends it to D. Thus light

ray travels from C to A to D at an angle with respect to horizontal in reference frame r1.Suppose

time flow during the ray reaches from C to D in reference frame r1 is Δt1 and the time flow during the

ray reaches from C to D in reference frame r2 is Δt2. Suppose speed of light in reference frame r1 is c.

Now,

(31)

(32)

Now, for reference frame r1:

(33)

Now, for reference frame r2:

For reference frame r2, horizontal distance y and light path x increases due to the motion of

reference frame r1 with respect to reference frame r2. From Δ ABC (or Δ ABD), the increase in the

horizontal distance can be found by considering the time interval equal to the time taken for ray to

reach from C to A (or A to D) which is equal to

for reference frame r2. Therefore increase in the

horizontal distance

(34)

Here we will assume that speed of light can be different for different reference frames and simple

Galilean velocity addition is true for speed of light also.

Using Galilean velocity addition, speed of light for reference frame r2 can be obtained as

speed of light in r2 √ (35)

√

(36)

Applying Pythagorean relation for reference frame r2, we get

(37)

From equations (31), (34), (36) and (37), we get

(

)

(

)

cot

( cos sin )

cot

( cos sin )

cosec

cosec

cosec

(

sin )

(

)

From equation(33) putting

Dividing both sides by , we get

(

)

(

)

(38)

Above mentioned equation (38) can be considered a quadratic equation in variable (

). We will try

to solve it using equation √

. Here

and

√

or

But

is not acceptable solution in reality. One reason for rejection can be given

like this: Time always flows in positive direction. In other words, amount of time always keeps on

increasing. Hence and have always positive values. Therefore

must have positive value.

But

gives negative value for some range of and . Another reason can be given

like this:

will have different value for different values of and . Even for a particular value of

characterizing a reference frame, there can be multiple values (possibly infinite) of , giving multiple

values (possibly infinite) of ratio

. Hence for a particular reference frame having speed will have

multiple values (possibly infinite) of ratio

because there can be multiple values (possibly infinite)

of . But this is not possible. It is completely against our common sense and what we observe in

reality. Therefore

is not acceptable solution.

Hence only one solution is acceptable from the two solutions obtained above. The acceptable

solution is

This solution obtained above that

is also consistent with intuitive common sense. It also

suggests that time flow in both reference frames remain same and time dilation does not exist.

In the derivation of the result obtained above, simple Galilean velocity addition was used for speed

of light and the result obtained by using that Galilean velocity addition is consistent with intuitive

common sense. Hence above analysis verifies and validates that simple Galilean velocity addition is

true for the speed of light also. It also validates the fact that speed of light can be different in

different reference frames.

Conclusions:

Galilean velocity addition works for speed of light also

Speed of light can be different in different reference frames

The fundamental assumption in special theory of relativity that speed of light is same in all

reference frames is wrong

Special theory of relativity is wrong

Time flow in both reference frame described above is same. This result can be extended to

all reference frames and we can say

Time flow in all reference frames is same and time dilation does not exist

Author: Lalit Vadher E-mail: lalitvadher@gmail.com