We apply force 1 Newton in X-direction & 1 N in Y direction on any moving substance.

Then by **special theory of relativity**, actual forces get acted on substance are (1+a) N & (1+ N.

This shows that something is seriously wrong in relativity. Where applied forces is different than acting forces.

I am putting mathematics of SR here

STEP 1:-This problem can easily be understood by following paradox.

{Before starting this paradox, I want to put one relativity formula’s given in any standard book of relativity for example “Page no. 135 of Elements of special relativity” by Dr T.M. Karade, Dr K S Adhav & Dr Maya S Bendre.

In any frame, for force in X-direction by S.R.

F_{x} = d/dt( y. mo. u_{x}) where y=(1-u^{2}/c^{2})^{-0.5}

So, after differentiation

F_{x}= y. mo. (du_{x}/dt) + y^{3}. m_{o}. {u_{x}/c^{2}}. (u . du/dt)

F_{x}= y. mo. a_{x} + y^{3}. mo. {u_{x}/c^{2}}. (u . a) -----(A)

We know, u^{2}=u_{x}^{2}+u_{y}^{2}+u_{z}^{2}

So, after differentiation

2 u. (du/dt) = 2.u_{x} (du_{x}/dt) +2.u_{y} (du_{y}/dt) + 2.u_{z} (du_{z}/dt)

2 u. a = 2.u_{x} a_{x} +2.u_{y} a_{y} + 2.u_{z} a_{z}

u. a = u_{x} a_{x} + u_{y} a_{y} + u_{z} a_{z} --------(

from (A) & (

So, F_{x}=y. mo. a_{x}+y^{3} mo. (u_{x}/c^{2}} (u_{x} a_{x}+u_{y} a_{y}+u_{z} a_{z}) ------(1)

This is general equation of force in X-direction in S.R.

**Now, Paradox**:-

On frictionless platform, object is moving with constant velocity u_{x} in X-direction & only magnetic force is acting in Y-direction & there is **acceleration in Y-direction only with velocity u**_{y}**.**

(& F_{z}=0)

If we apply eq(1) to this case then result will be

F_{x}= y^{3} mo. (u_{x}/c^{2}} u_{y} a_{y} ---------- as a_{x}=0

Or F_{x}=F_{ay} as this force is form due to ‘a_{y}’ & ‘u_{y}’ only

Mean’s even there is **no magnetic force** acting on object from outside in x-direction & no ‘a_{x}’ then also **above force will act on object in +ve direction of x-axis due to ‘a**_{y}**’ **

**Important point (1):-**

**Mean’s applied magnetic force on object in X-direction is 0 & acting force in X-direction is ** Fx= y

^{3}mo. (u

_{x}/c

^{2}} u

_{y}a

_{y}+0 or F

_{ay}+0=F

_{ay}

**-------------------------------------------------------------------------------------------------------------------------------**

**STEP 2:-Now, Force acting in X-direction is ** F_{x}= y^{3} mo. (u_{x}/c^{2}} u_{y} a_{y} or F_{ay}

**Now, after this happen, very small magnetic force of same intensity**

** -f**_{x}** = -**y^{3} mo. (u_{x}/c^{2}} u_{y} a_{y} or -F_{ay} start **acting on object in direction opposite to above force (but velocity is still positive u**_{x}**) & cancel that above force. **

**Mean’s equation (1) becomes**

0=y. mo. a_{x}+y^{3} mo. {u_{x}/c^{2}} (u_{x} a_{x}+u_{y} a_{y})

Or 0 =y. mo a_{x}. (1+ y^{2} {u_{x}^{2}/c^{2}} ) + y^{3} mo. (u_{x}/c^{2}} u_{y} a_{y} =y. mo a_{x}. (1+ y^{2} {u_{x}^{2}/c^{2}} ) +F_{ay}

(Here as F_{ay}= y^{3} mo. (u_{x}/c^{2}} u_{y} a_{y})

Mean’s F_{ay} = y. mo. -a_{x}. (1+ y^{2}. {u_{x}^{2}/c^{2}} )

**Mean’s there must be acceleration in –ve X-direction to fulfill above equation of S.R.**

**Now, see above equation carefully, it is of nature**

** 0= -f**_{x}** + F**_{ay}

**Important point (2):- Mean’s applied magnetic force on object in X-direction is -f**

_{x}**&**

__acting force__in X-direction is -f

_{x}**+ F**

_{ay}**= 0 or 0.**

**Here, resultant force in X-direction is zero but there is acceleration.**

**STEP3:- same things happen for +ve force in X-direction (for less than F**_{ay}** or more)**

Now, I am generalizing above result.

Step 1 & 2 clearly shows that when we apply any magnetic force (F_{mx}) in X-direction on the object, actual force acting on object is more & that quantity is (F_{mx}+F_{ay})

Similarly,

If we apply any magnetic force (F_{my}) in Y-direction on the object then actual force acting on object is more & that quantity is (F_{my}+F_{ax})

This is completely complicated results, which says that applied force & acting forces on objects are different & more in S.R.

**STEP4:- Force does work, consume energy, gain energy & we must know that energy cannot be created. It can be transferred only:-**

From above setup it must be clear that energy get transfer from magnet to object but if applied force is less than acting force then energy gain by object will be more than energy loose by the magnet. Means due to more work done by more force for same displacement, more energy get generated.

HERE, more energy (& force) is the problem.

**Where this additional energy (or force) does comes from?**

**There is no answer in S.R. for this problem.**

#### Attached Files

**Edited by maheshkhati, 12 September 2017 - 07:20 AM.**