Top page (correct Bohr model )

Strange "spin" is NOT a real thing

Special relativity is false.

- Relativistic momentum causes paradox.
- Observer can rotate lever without touching it !
- Transformation of Newtonian forces.
- Solution to this paradox depends on "fake" center !

*(Fig.1) Four momentum. *

Special relativity claims Four-momentum (= E/c, p ) transform like spacetime ( ct, x, y, z ) under Lorentz transformation.

"**E**" and "**p**" are relativistic momentum and energy, respectively.

Newtonian **force F** is defined as the rate of change of **momentum** (= p ) with respect to time.

And the time derivative of energy (= E ) is equal to force F × velocity (= u ).

*(Fig.2) Newtonian force F _{y} is reduced when it is moving.*

For relativistic energy and momentum to satisfy the equations of Fig.1, force F_{y} must **decrease** when it is moving.

See various sites ( this (v=0), this (p.3), this (p.4,5), this (p.15) ).

In Fig.2 left, an observer (= K ) and an object (= M ) are at rest, and force **F _{y}** is applied to this object in y direction.

When the observer starts to move in x direction at velocity "v", the object is **moving** in the opposite direction from his viewpoint (= K', Fig.2 right ).

Force in **y** direction in K' frame is **reduced** according to special relativity.

*(Fig.3) Two equal forces F _{y} are in equilibrium.*

In Fig.3, two equal forces F_{y} are applied to the **stationary** lever in K frame (= observer at rest ).

This lever has two equal arms on both sides of a fulcrum, so it is in **balance**, NOT rotate in K frame.

One of these forces (= Fig.3 left ) is **at rest**, and another force ( source ) is **moving** at velocity "v" in x direction.

*(Fig.4) Lever rotates from the viewpoint of moving observer (= K' ) ? *

When the observer is moving at "v" in x direction (= K' ), this lever and left force start to **move** in the opposite direction.

As a result, this force (= Fig.4 left ) is **reduced**, as shown in Fig.2.

On the other hand, right force looks **stationary** in K' frame, it is **increased** by observer's movement.

As a result, this lever **rotates** clockwise only in K' frame in special relativity !

This unreasonable transformation of force F_{y} is one of causes of right angle lever paradox below.

Of course, this paradox has **NO** solutions, which means relativistic momentum and energy are **wrong**.

*(Fig.5) Special relativity gives observer "supernatural power" ? *

Here is a right-angle lever with two arms which can rotate around a stationary point.

In Fig.5 left, two arms of this lever are **equal** in the length.

And the forces F acting on them are completely the **same**, which directions are perpendicular to each other.

So in Fig.5 left, this lever is **at rest** ( not rotating ) due to **equilibrium** of forces. ( torque: F×L = F×L. )

But when the observer starts to move, this lever starts to **rotate**, though he doesn't touch it ! (= Fig.5 right. )

This is clearly a fatal **paradox**, and shows special relativity is **wrong**.

*(Fig.6) Why the lever "rotates" just by observer's movement ? *

Here we explain why this right-angle lever rotates just by simple observer's movement.

One arm (A-2) is Lorentz-contracted in the moving direction. ( = x direction )

And furthermore, the force F, which is **perpendicular** to the moving direction, becomes **weaker** than the orginal F according to the special relativity, by a factor of (1-β^{2})^{1/2}.

As a result, torque of the lever is **NOT** zero **only** from the viewpoint of moving observer.

From the stationary observer, the lever torque is **zero**. This is paradox.

*(Fig.7) Trasformation of Newtonian forces in x, y directions. *

The force in x direction remains the same ( **F' _{x} = F_{x}** ), and the force in y direction is reduced (

In Fig.7, Lorentz transformation is in x direction.

And "*rest*" means the force applied to a object at **rest**.

*(Fig.8) Force parallel to observer's movement is invariant ( F _{x} = F'_{x} )*

In Fig.8, an object (= M ) is **moving** at velocity **u** = "**v**" in x direction relative to the stationary observer in K frame.

Force **F _{x}** (= ordinary force, NOT four-force ) is applied to this object in K.

As seen by the observer **moving** at velocity "v" in x direction (= K' ), this object looks **stationary**.

So, **K'** frame is the **rest** frame of the object.

The force F_{x} transforms to F'_{x} in K' frame.

According to special relativity, these forces are equal ( **F _{x} = F'_{x}** ) in any frames.

*(Fig.9) Force perpendicular to observer's movement decreases ( F _{y} < F'_{y} )*

On the other hand, force (= F_{y} ) perpendicular to the observer's movement is **reduced** in K frame than the force (= F'_{y} ) in the rest frame (= K' ) of the object.

As seen by the **moving** observer in K' frame, the object is **at rest**, and force F'_{y} is applied (= Fig.9 right ).

From the perspective of stationary observer in K frame, the object is **moving**, and its force decreases ( **F _{y} < F'_{y}** ).

*(Eq.1) Lorentz transformation of space-time coordinate.*

According to special relativity, the space-time coodinate (= t, x, y, z ) transform, obeying Lorentz transformation.

*(Eq.2)*

Special relativity claims Four-momentum transform like spacetime ( ct, x, y, z ) under Lorentz transformation. See this (p.15) and this (p.12)

This transformation of relativistic energy (= E ) and momentum (= p ) is used in Dirac equation, as shown on this page.

Above transformation of Newtonian force is **indispensable** for this four-momentum relation.

*(Eq.3) *

So, the momentum (= p'_{x} ) in K' frame is given by Eq.3.

Relativistic momentum (= p_{x} ) in x direction and energy (= E ) in K are **mixed**.

*(Eq.4) From the viewpoint of "stationary" observer (= K ).*

From the viewpoint of **stationary** observer (= K ), an obejct is moving at velocity **u** (= v ) to the right.

Force (= F ) is equal to the rate of change in momentum (= p ).

The rate of change in energy (= E ) is equal to the force × velocity "u".

Because the force × moving distance is energy E.

*(Eq.5) From the viewpoint of moving observer (= K' )*

In K' frame, the observer is moving at velocity "v" to the right.

The object is moving in the same direction, so it looks stationary as seen by the observer in K'.

So the velocity of the object in K' is **u' = 0**.

The momentum (= p' ) and energy (= E' ) are given by the relation of Eq.3.

About the calculation of Lorentz transformation of forces in each direction, see this section

*(Eq.6) Lever is at rest in K' frame, but rotates only in K frame.*

A right angle lever is moving at velocity u = v in x direction.

The observer in K' frame is also moving at the same velocity v in x direction.

*(Eq.7) Lever is at rest in K' frame (= moving observer ).*

So this lever is **stationary** in K' frame (= rest frame of the lever ).

We suppose the lengths of the lever arms and the applied forces are equal in K' frame.

As a result, the lever is in equilibrium, and does **NOT** ratate (= torque is **zero** ) in K' frame.

*(Eq.8) Lever rotates ONLY in K frame. Torque in K frame is NOT zero.*

From the perspective of stationary observer in K frame, this lever is moving to the right.

The lever arm parallel to the moving direction is Lorentz **contracted**.

And the Newtonian force F_{y} is **reduced**, compared to F'_{y} at rest frame.

As a result, torque τ is **NOT** zero, as shown in Eq.8.

*(Fig.10) Strange energy flows inside lever ? Virtual center of rotation !?*

To solve this right-angle lever paradox, they introduced **unrealistic** concepts such as "**virtual**" center of rotation, and energy **flow**, as shown on this site and this site.

"p" is also "virtual" momentum flowing through the lever.

They suppose virtual center of rotation exists at the point of observer.

*(Fig.11) "Virtual" torque is NOT real.*

Torque is given by the force times radius.

But they changed this definition, as shown in Fig.11 lower and this page.

They introduced **virtual** center of rotation at oberver.

So this ( virtual ) radius "r" inreases as **r = vt** ( "v" is velocity ).

Torque can be expressed by derivative of this radius "r" × fake momentum p.

In Fig.10, the lever is moving at speed "v" with respect to K frame in the x direction.

So they supposed the external force F_{x} do work "**Fv**" per second on the point "**1**" of the lever.

*(Eq.9) ↓ Virtual energy flow.*

And they supposed this **strange** energy **flows** to the point "**A**" at the speed of "**U**" inside the lever.

In Eq.9, **T** is the time needed for this energy to reach the point "A".

And **E** is the **accumulated** energy during this time T inside the whole arm of "1-A" .

*(Eq.10) ↓ "Virtual" accumulated energy inside arm.*

Using the relation between relativistic energy "E" and momentum "p", Eq.10 is the **momentum** accumulated in the arm "1-A".

( Relativistic momentum p = energy E × U/c^{2}. U is velocity of virtual energy. )

And they **artificially** supposed this energy flows **outside** from the pivot "**A**" due to the law of action and reaction.

*(Eq.11) ↓ "Fake" torque.*

And again they used "ad-hoc" idea that the observer of K frame is the **center of rotation**, though this observer is **NOT** connected to this lever arm by any string or forces.

They utilize the fact that torque is given by the time derivative of the angular momentum.

But they artificially **replaced** the meaning of the torque by other **unreal** concepts, as shown in Fig.11.

*(Fig.12) Torque based on "virtual" center coincides with "real" torque !?*

The distance between the lever and observer is "**vt**", so the new angular monentum is L = "**vtp**".

So the time derivative of this L becomes "**vp**", they insist.

As a result, the torque τ = "**vp**" (= Eq.11 ) becomes just equal to Eq.8, and they insisted the lever torque is used in the strange energy flow and no rotation happens.

*(Fig.13) Force F does NOT do work on point 1.*

But this method depends on some **unreasonable** assumptions.

First, force F_{x} does **NOT** do work on the point "1", when the lever torque is zero ( or weakened ).

( This force F_{x} is **cancelled** out by the force F_{y} because of **rigid** lever. )

For example, we suppose the lever at rest is in equilibrium due to equal forces (= F ) applied to two arms.

( In this case, force F does **NOT** work on this lever, because the lever is **neither** moving nor rotating. )

But, when the observer starts to move to the left, this lever starts to move to the right from the observer's **perspective**.

Based on the above "ad-hoc" assumption, this force F **does** work just by observer's movement !

Of course, it is **impossible**.

*(Fig.14) Lever rotates "and" does NOT rotate at the same time ? ← Paradox ! *

And as you notice, this **energy flow** inside the lever and the strange center of rotation are **NOT** true.

In conclusion, right-angle lever paradox has **NOT** been solved and shows the relativistic theories including QED, and standard moded are all wrong.

*(Eq.12) Relativistic momentum p.*

In this paper, they tried to solve this right angle lever paradox using **different** relativistic mass in x, y directions.

Suppose a object with rest mass (= m_{0} ) is moving at velocity u_{x} in x direction.

So the relativistic momentum is given by Eq.12.

*(Eq.13) Relativistic mass in x direction*

Using the momentum of Eq.12, we obtain the relativistic mass in **x** direction, as shown in Eq.13.

Here we use

*(Eq.14)*

In the same say, the relativistic mass in **y** direction becomes

*(Eq.15) Relativistic mass in y direction.*

From Eq.13 and Eq.15, you find the relativistic mass in x direction is **heavier** than that in y direction.

*(Eq.16) Mass in x direction is heavier than y direction.*

From Eq.13, the acceleration (= a_{x} ) in the x direction is

*(Eq.17)*

From Eq.15, the acceleration (= a_{y} ) in the y direction is

*(Eq.18)*

So the acceleration is **bigger** in y direction than x direction due to the **difference** of relativistic masses between x and y directions.

*(Eq.19) Torque = acceleration ( instead of force ) × length ?*

They use acceleration instead of force, when calculating torque.

From Eq.17, Eq.18, Eq.19, the torque in K frame (= lever moving frame ) becomes

*(Eq.20)*

As a result, also in K frame, the torque is zero. So there is no right-angle lever paradox, they insist.

But is it really so ?

Eq.17 and Eq.18 use the **difference between the relativistic masses** in the x and y direction.

The relativistic mass in the x direction is heavier than that in the y direction,
so the acceleration is bigger in the y direction.

This cancels out the torque ?

In fact, the relativistic masses and accelerations are the **same** in both cases, because of "**rigid**" body.

*(Fig.15) Relativistic masses are " equal " in both cases.*

In Fig.15 left, the force **F y** is trying to rotate the whole lever

In this case, the

So this rotation

In Fig.9 right, the force **F _{x}** is trying to rotate the whole lever

As a result, **both** these two rotations ( due to F_{x} and F_{y} ) contain **same** relativistic mass.

So the torque = "force × length" is **equal** to "acceleration × length", which situation does **NOT** change.

This means idea of acceleration **cannot** stop rotation of this lever.

This right-angle lever **remains** a fatal paradox of special relativity.

*(Ap.1) K' and object are moving at v (= u ) in x direction.*

K' frame and the object are moving at velocity **u = v** in x direction, relative to K frame.

Momentum and energy of the object as seen by observers in K and K' frames are ( p, E ) and ( p', E' ), respectively

Defferentiating p'_{x} (= momentum in K' ) with respect to the time t (= K ), we get Ap.1.

Here we use transformation of Ap.2 and the definition of Ap.3.

*(Ap.2) Lorentz transformation of four-momentum.*

*(Ap.3) From the viewpoint of different observers ( K and K' ).*

Using Lorentz transformation of the time, and differentiating t' with respect to t, we obtain

*(Ap.4) *

"u_{x}" is the object's velocity in x direction from the viewpoint of K frame.

Using Ap.4 and comparing the result with Ap.1, we get

*(Ap.5)*

Here, the object is moving in x direction at **u = v** in K frame, so we get the relation of

*(Ap.6) Force in x direction does NOT change. ( F _{x} = F'_{x} )*

F'_{x} is the force in x direction acting on the object in K' frame (= rest frame of the object ).

According to the special relativity, force in x direction is **invariant** in any frames. ( **F' _{x} = F_{x}** )

*(Ap.7) Transformation of force in y direction.*

*(Ap.8) Force in y direction is reduced. ( F' _{y} > F_{y} )*

As shown in Ap.8, the force (= F_{y} ) in y direction is **reduced** compared to force (= F'_{y} ) of rest frame.

This is one of reasons why right angle lever paradox happens.

*(Ap.9) Ordinary velocity (= u ) from stationary observer in K frame.*

In this section, we explain transformation of forces with respect to relativistic **four-forces**.

In Ap.9, an object is moving at velocity "u" relative to the stationary observer (= K frame ).

*(Ap.10) Lorentz invariant proper time (= τ )*

In relativistic kinematics, we often use the convenient proper time, which is Lorentz invariant.

So, this proper time is **NOT** real "time" in each frame.

They utilize this proper time ONLY for making equation of motion Lorentz-covariant.

That's all.

*(Ap.11) *

From Ap.11, we can prove this proper time τ has a Lorentz-invariant form.

See also this page.

From Ap.11, we have

*(Ap.12) *

In relativistic world, the **ordinary** velocity (= u ) transforms very complexly.

Because, both the time (= dt ) and space (= dx ) coordinates change.

So they tried to use convenient **four-velocity** (= ω ), as shown on this (p.111) and this site.

*(Ap.13) Ordinary velocity (= u ) and four-velocity (= ω )*

The denominator of this four-velocity ω is Lorentz-invariant proper time.

So, only its numerator (= dx, dy, dz, cdt ) transforms under Lorentz transformation.

*(Ap.14) 0-component of this four-velocity*

So subsituting "ct" and Ap.11 into Ap.14, we can get the zeroth-component of this four-velocity.

*(Ap.15) Relativistic equation of motion.*

They define relativistic equation of motion like Ap.15.

The left side is derivative of four velocity with respect to proper time.

So again, this transforms like four-vector. See this (p.7) and this site.

The right side (= f ) of Ap.15 is called "**four-force**", very artificial concept.

*(Ap.16) Four-force f, ordinary force F.*

On this page, "**F**" denotes ordinary ( **Newtonian** ) force, and "f" denotes four (vector) force.

This four-force transforms like four-vector, which is **NOT** a real force.

Using the proper time of Ap.10, we have

*(Ap.17) *

Substituting Ap.13, Ap.16 and Ap.17 into the relativistic equation of motion (= Ap.15 ), we get the relation of Ap.18.

*(Ap.18) *

If we define relativistic **momentum p** as

*(Ap.19) *

Ap.18 becomes

*(Ap.20) *

Ap.20 is **ordinary** equation of motion.

Only "p" must be relativistic momentum like Ap.19.

*(Ap.21) *

For these four-forces to transform as four-vector, 0-component of four-vector must have the form of Ap.21.

Substituting Ap.14 and Ap.21 into Ap.15, we have

*(Ap.22) *

Here we define relativistic energy of

*(Ap.23) *

An object is moving at velocity u = v in x direction in K frame, so

*(Ap.24) *

Substituting Ap.24 into Ap.16 and Ap.21, we have

*(Ap.25) *

Four-force transform like four-vector ( ct, x, y, z ) under Lorentz transformation.

Using 0 and x components of Ap.25, we obtain

*(Ap.26) *

In K' frame, the object is at rest, so the velocity (= u' ) in K' is **zero**.

So, four-force in K' frame is **equal** to the ordinary Newtonian force.

*(Ap.27) *

From Ap.26 and Ap.27, we find Newtonian force in x direction is **invariant** under Lorentz transformation.

On the other hand, four-vector force f_{y} **doesn't** change under Lorentz transformation, so

*(Ap.28) *

We can get the same result of Ap.8.

2014/6/1 updated. Feel free to link to this site.