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 assumption. 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 r

_{2}. Suppose a co-ordinate system is fixed in reference frame r_{1}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 r_{1}in which the source resides are moving with respect to another reference frame r_{2}and this motion of reference frame r_{1}with respect to reference frame r_{2}is in positive x-direction with speed v or in other words source is moving with speed v with respect to reference frame r_{2}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 r_{1}.**Therefore for co-ordinate system fixed in reference frame r**

_{1}:Using Maxwell’s equation, we will find

**B**.Therefore from Maxwell’s equation

Integrating and finding

**B,**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 phase 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.

Differentiating it, we get

**But for reference frame**

**r**

_{2}**the distance in x direction increases by the amount**because of motion of reference frame r

_{1}with respect to reference frame r

_{2}in positive x-direction with speed v. Therefore denoting the x-co-ordinate of the co-ordinate system fixed in r

_{2}as and assuming when, we get

**Therefore to get the velocity of wave with respect to reference frame**

**r**From equation (1.a) and (1.c), we get

_{2}:

Differentiating it to get phase velocity with respect to reference frame r

_{2}, we get**For reference frame r**(Using the co-ordinate system fixed in r

_{1}:_{1}as described before)

From equations (5) & (7)

From equations (2) & (7)

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

From equations (4) and (7)

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

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

**But speed of light with respect to reference frame r**

_{2}, using equation (6) and (16):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 but 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 r

_{1}as shown in figure just above and reference frame r_{1}is travelling with respect to reference frame r_{2}with speed v in horizontal direction. In reference frame r_{1}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 r

_{1}, 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,Suppose for reference frame r

_{2}, total time flow during the ray’s travel from A to M is In reference frame r_{2}, 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 r_{1}with respect to reference frame r_{2}. In other words, the light ray will travel some horizontal distance due to horizontal speed v of reference frame r_{1}with respect to reference frame r_{2}and vertical distance h in reference frame r_{2}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.

From (18) and (19)

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**:

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

v 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 r

_{1}) is perpendicular to the direction in which reference frame of source (reference frame r_{1}) travels with respect to another reference frame (reference frame r_{2}). 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 r_{1}when reference frame r_{1}is travelling in horizontal direction with respect to reference frame r_{2}. 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 r_{1}(with respect to reference frame r_{1}) and reference frame r_{1}is assumed to travel in horizontal direction with respect to reference frame r_{2}.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 r

_{1 }travels at speed v with respect to reference frame r_{2}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 q with respect to horizontal in reference frame r_{1}.Suppose time flow during the ray reaches from C to D in reference frame r_{1}is Δt_{1}and the time flow during the ray reaches from C to D in reference frame r_{2 }is Δt_{2}.Now, for reference frame r

_{1}:Now, for reference frame r

_{2}:For reference frame r

_{2}, horizontal distance y and light path x increases due to the motion of reference frame r_{1}with respect to reference frame r_{2}. 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 r_{2}. Therefore increase in the horizontal distanceApplying Pythagorean relation for reference frame r

_{2}, we getFrom equations (21),(24) ,(25)and (26), we get

Now,1+cot

^{2}θ=cosec^{2}θThe above equation can be considered a quadratic equation of variable . Solving for it, using equation

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.

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=mc

^{2 }equation was derived which should also be wrong because of the wrong base from which it was derived. The formal proof that E=mc^{2}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

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)

Squaring both sides and rearranging we get,

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=forcedistance. Treating work done and energy as same quantity

From equation (B) and (D)

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 is 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 r

_{1 }travels at speed v with respect to reference frame r_{2}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 q with respect to horizontal in reference frame r_{1}.Suppose time flow during the ray reaches from C to D in reference frame r_{1}is Δt_{1}and the time flow during the ray reaches from C to D in reference frame r_{2 }is Δt_{2}. Suppose speed of light in reference frame r_{1}is c.Now, for reference frame r

_{1}:Now, for reference frame r

_{2}:For reference frame r

_{2}, horizontal distance y and light path x increases due to the motion of reference frame r_{1}with respect to reference frame r_{2}. 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 r_{2}. Therefore increase in the horizontal distance**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 r

_{2}can be obtained asApplying Pythagorean relation for reference frame r

_{2}, we getFrom equations (31), (34), (36) and (37), we get

Above mentioned equation (38) can be considered a quadratic equation in variable. We will try to solve it using equation.

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