Friday, January 7, 2011

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 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 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:
Assumeclip_image002, clip_image004 andclip_image006                 (1)
Using Maxwell’s equation, we will find B.  
   clip_image008                                                                    (2)
Finding clip_image010 from assumed values of Ex, Ey, and Ez, we get
clip_image012                                    (3)

Therefore from Maxwell’s equation
clip_image014                                             (4)

Integrating and finding B,                         
   B clip_image016(0, 0,clip_image018)                                                            (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 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.
With respect to reference frame r1:clip_image020
clip_image022                                                           (1.a)
Differentiating it, we get 
clip_image024
clip_image026Propagation speed with respect to reference frame r1clip_image028                  (1.b)
But for reference frame r2 the distance in x direction increases by the amountclip_image030 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 asclip_image032 and assumingclip_image034 whenclip_image036, we get
clip_image038                                                         (1.c)
Therefore to get the velocity of wave with respect to reference frame r2: From equation (1.a) and (1.c), we get
clip_image040
Differentiating it to get phase velocity with respect to reference frame r2, we get
clip_image042
clip_image044          (6)
Now, we haveclip_image046
clip_image048                                                                (7)
For reference frame r1: (Using the co-ordinate system fixed in r1 as described before)
From equations (5) & (7)                          
 clip_image050                                                              (8)
Finding clip_image052 from equation (8), we get
clip_image054                                  (9)
From equations (2) & (7)
 clip_image056   clip_image058                             (10)
Fromclip_image060 and equation (7)
                                          clip_image062                                 (11)
Multiplying equation (10) and (11), we get,
clip_image064                                                 (12)
From equations (4) and (7)                
          clip_image066                                                              (13)
Using equations (1), (3), (9), and (13) for putting values into equation (12), we get                                                                
clip_image068                                           (14)

clip_image070
clip_image072                                                                 (15)
From equation (1.b) and (15),
clip_image074               (16)
But speed of light with respect to reference frame r2, using equation (6) and (16):
clip_image076   (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 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
clip_image078
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:
clip_image080
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 isclip_image082. 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,
clip_image084                                                                         (18)
Suppose for reference frame r2, total time flow during the ray’s travel from A to M is clip_image086 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 clip_image088 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.
clip_image090
Here also according to the wrong assumption assumed by Einstein, the speed of light should be c.
In clip_image092 time the light ray travels AM distance.
clip_image094                                                           (19)
From (18) and (19)
clip_image096
clip_image098
clip_image100
clip_image102
clip_image104
clip_image106                                                                   (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:
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 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 angleclip_image108. By denoting the direction using a variableclip_image108[1], we can obtain general result in terms ofclip_image108[2]. And by putting specific value of clip_image110 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 clip_image112 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 q 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.
clip_image114
Now,clip_image116                                                       (21)
clip_image118                                                        (22)
Now, for reference frame r1:
clip_image120                                                                  (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 toclip_image122 for reference frame r2. Therefore increase in the horizontal distance
clip_image124                                                                       (24)
And we haveclip_image126                                      (25)
Applying Pythagorean relation for reference frame r2, we get
clip_image128                                                      (26)
From equations (21),(24) ,(25)and (26), we get
clip_image130 
clip_image132
clip_image134
clip_image136
clip_image138
Now,1+cot2θ=cosec2θ
clip_image140
clip_image142
From equation(23) puttingclip_image144
clip_image146
clip_image148
clip_image150                                              (27)
The above equation can be considered a quadratic equation of variable clip_image152. Solving for it, using equationclip_image154
clip_image156                                                     (A)
This equation shows that for different values of θ and v, there can be different values of the ratio clip_image152[1]. 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 clip_image158 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 onlyclip_image160 rad) in Einstein’s special theory of relativity was wrong. In wrong interpretation of time dilation in special theory of relativity,  it was derived thatclip_image162 which was the case obtained for clip_image158[1] rad only and it disregarded the other infinite possible values ofclip_image108[3] 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,
clip_image164
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

clip_image166                                                (28)


which again depends on θ and does not make sense.
From above mentioned equation (28), if θclip_image168  radians (which can always happen in infinite number of ways)
clip_image170                                                                     (29)

Squaring both sides and rearranging we get,

clip_image172                                                       (30)
Here clip_image174 and c are assumed constants.Doing differentiation, we get
clip_image176
clip_image178
clip_image180    (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=forceclip_image182distance. Treating work done and energy as same quantity                                        
energy clip_image184andclip_image186
clip_image188                                                           (C)
Puttingclip_image190  in equation (C) to convert formula in clip_image192 to formula in clip_image194, we get
clip_image196                                                   (D)
From equation (B) and (D)           
clip_image198
clip_image200                                                                    (E)
Integrating both sides                                  
clip_image202
Suppose clip_image204, then differentiating it , we getclip_image206 which is completely opposite to equation (E). Therefore clip_image208. Similarly suppose clip_image210 then differentiating it, we get clip_image212 which is also completely opposite to equation (E). Hence clip_image214. 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 clip_image216sign instead of clip_image218 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 assumingclip_image160[1] 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 clip_image158[2] 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 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 q 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.
clip_image114[1]
Now,clip_image116[1]                                                     (31)
clip_image118[1]                                                      (32)
Now, for reference frame r1:
clip_image120[1]                                                               (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 toclip_image122[1] for reference frame r2. Therefore increase in the horizontal distance
clip_image124[1]                                                                    (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
clip_image220                                       (35)
clip_image222                                 (36)
Applying Pythagorean relation for reference frame r2, we get
clip_image128[1]                                                       (37)
From equations (31), (34), (36) and (37), we get
clip_image224
clip_image226
clip_image228
clip_image230
clip_image232
clip_image234
clip_image236
clip_image238
From equation(33) puttingclip_image144[1]
clip_image240
clip_image242
clip_image244
Dividing both sides byclip_image246, we get
clip_image248
clip_image250                                                (38)
Above mentioned equation (38) can be considered a quadratic equation in variableclip_image252. We will try to solve it using equationclip_image154[1].
Here clip_image254
clip_image256
clip_image258
But clip_image260 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 clip_image262 and clip_image092[1] have always positive values. Therefore clip_image264 must have positive value. But clip_image260[1] gives negative value for some range of clip_image112[1] andclip_image266. Another reason can be given like this: clip_image264[1] will have different value for different values of clip_image112[2] andclip_image266[1]. Even for a particular value of clip_image268 characterizing a reference frame, there can be multiple values (possibly infinite) ofclip_image108[4], giving multiple values (possibly infinite) of ratioclip_image270. Hence for a particular reference frame having speed clip_image268[1] will have multiple values (possibly infinite) of ratio clip_image264[2] because there can be multiple values (possibly infinite) ofclip_image108[5]. But this is not possible. It is completely against our common sense and what we observe in reality. Therefore clip_image260[2] is not acceptable solution.
Hence only one solution is acceptable from the two solutions obtained above. The acceptable solution is
clip_image272

This solution obtained above that clip_image274 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