Relativistic Photon Moving with Velocity Greater than c

Libor Neumann

Prague, Czech Republic, email: LNeumann@chello.cz

The particular case is described where photon emitted from the moving light source is moving with the velocity greater than c in the stationary co-ordinate system. The photon velocity is dependent on the light source velocity. The velocity evaluation is strictly based on the special relativity theory transformation.

03.30.+p

The paper title sounds like contradiction.
The second relativity theory postulate is “Any ray of light moves in the
“stationary” system of co-ordinates with determined velocity *c*, whether
the ray be emitted by stationary or by moving body” [1]. Moreover the
A. Einstein original paper includes the proof of the special relativity theory
(SRT) transformation correspondence with the second postulate in §3.

Following text describes the particular case where properly used mathematical apparatus of SRT gives different result.

The following description uses definitions
and symbols from §3 of SRT original description ^{1}. Two Cartesian
co-ordinate systems (*K* - stationary system and *k* - moving one)
are used. The stationary system *K* includes axes *x,y,z* and time *t*.
Moving system *k* includes axes *x**,**h**,**z * and time *t*. The origin of the
moving system moves with the velocity *v *(in* x *axis direction). * *

We
use well known relativistic basic co-ordinate and time transformations
described in §3 ^{1}

_{}

where

_{}.

We
use velocity transformation equations described in §5 ^{1}. General
point or object moves with the velocity *w* (component and co-ordinate
system is distinguished by index).

_{}

SRT is based on the space co-ordinate
transformation and on the time transformation. We look for the particular
subset, where the time in stationary co-ordinate system *K* is equal to
the time in moving co-ordinate system *k*. The analysis could be simpler
if the subset exists.

_{} (1)

We can solve equation (1). We can see; the
plain orthogonal to the direction of the moving co-ordinate system velocity
exists, where time in both co-ordinate systems is equivalent. We name the plain
*ET *and we index the values connected with the plain by *ET*.

Let us analyse the features of the
particular plain* ET*. We discover:

*ET*plain position computed in moving co-ordinate system*k*based on the same idea as above is described by (2) i.e. by the same equation with opposite sign as (1). It respects opposite orientation of the velocity*v*.

_{} (2)

- We can verify
the relation between (1) and (2) by the standard relativistic co-ordinate
transformation of any point at
*ET*plain from stationary co-ordinate system*K*into moving co-ordinate system*k*(3) and vice versa.

_{} (3)

We gain the same result as (2).

*ET*plain moves in both co-ordinate systems. The velocity of*ET*plain in stationary co-ordinate system*K*in*x*direction is*x*Using (1) we gain_{ET}/t.

_{}

By SRT velocity
transformation of any point of *ET *plain velocity we gain

_{}

therefore

_{} (4)

It looks like symmetry expected in SRT. We gain the same results from (2).

- By the velocity transformation in orthogonal direction we gain equivalence (5).

_{} (5)

We can describe the item *c ^{2}/v^{2}*
with use of the

_{} (6)

Using (6) in (1) *ET *plain position
can be described by (7).

_{} (7)

We can see from (7) that *ET *plain is
exactly in the middle of the distance between the origins of the co-ordinate
systems only for *b**=1* i.e. only for *v=0*. The
distance of *ET *plain from the stationary co-ordinate system *K*
origin is greater than *0.5 vt* for any *v>0 *(and *t**>0*).

We know from basic course of physics that
it is impossible to add or compare physical quantity without measuring unit
knowledge and without conversion of all quantities into the common units. It
is impossible to add distance in meters with distance in feet. SRT ^{1}
does not solve the question of the measuring units in detail.

The detail analysis of SRT transformations from the measuring unit point of view gives surprising result.

We can find that measuring unit of the *y*
axis must be equal to the measuring unit of the *h* axis (due to equivalence of values and equivalence of distances).
The same result we gain in case of the *z* and *z* axes.

Similar situation is in case of time
measuring units at *ET *plain. The equivalence of time values (at *ET *plain)
(1), equivalence of the velocity values in orthogonal direction (5) and equal
measuring units of *y,z* and *h**,**z* axes imply the equal time measuring units in both co-ordinate
systems (at *ET *plain).

Additionally we can define that the
measuring units of *x,y*,z axes are equal (similarly to §3 ^{1}).

It is not possible to confirm the idea that
the measuring unit of *x* axis is
equal with the measuring unit of *x* axis as indirectly described in ^{1}.

Moreover obvious geometrical problem exists in case of this interpretation.

Figure1 - Distances between co-ordinate
systems origins and *ET* plain

Let us compare the sum of distances of *ET
*plain from the co-ordinate systems origins with the distance between the
origins. From the geometry we know, that the sum of distance between the
origin of the stationary co-ordinate system *K* and *ET *plain and
the distance of the same plain from the origin of the moving co-ordinate system
*k* must be equal to the distance between the origins.

The distance D between origins is from
the definition in §3 ^{1}

_{}

The same distance is described by (8) evaluated from (1), (2) (using (6)).

_{} (8)

We can see from (8), that equivalence of
distances can be valid only in case *b**=1* i.e. *v=0 *(if we consider the equal measuring units in both
co-ordinate systems).

**It means, if we consider the equal
measuring units of ***x*** axis
and x axis, SRT transformation can be valid only for v=0 (due to
the geometrical reasons connected with the co-ordinate systems origins
displacement). It gives no physical sense. **

We can try to solve the problem by
discovering the ratio between measuring units of the *x* and *x* axes.

We can suppose the ratio between measuring
units of *x* and *x* axes is *a*. We know
from (1),(2), and (4) that the original and transformed relations give the same
values (with opposite signs). We can describe the distance geometry with
respect of measuring unit ratio by (9)

_{} (9)

We can see from (9) that it is possible to
describe distance D equal *vt*
as sum the same values together with the fact that distance of *ET *plain
can be greater than *0.5 vt* in co-ordinate system *K* by the proper
selection of the measuring unit ratio.

We can compare (7) with (9). We can see
that **the geometrical problem of SRT transformation can be solved only if the
ratio of measuring units of ***x*** and x axes is **

It means, the physical length unit used for
the distance description at *x* axis is *b*-times
smaller than the length unit at the all other axes (*x,y,z,**h**,**z*). The same is valid for the velocity measuring units. We will
identify the velocity unit at the *x,y,z,**h**,**z** * axes by [U] and the velocity unit at the *x* axis [u] in the following text.

Let us have the light source moving with
velocity *v* in direction of *x* axis. It is placed in the origin of
the moving co-ordinate system *k* as usual in SRT. Both co-ordinate
systems origins are equal at *t=**t**=0*. The light source emits photon *P* at *t=**t**=0* with the velocity *c* (measured in units [U]) in the direction
that photon moves in *ET *plain in direction of *h* (*y*) axis. Photon moves in *x* axis direction with the velocity (10) measured in [u] – measuring
unit of *x *axis.

_{} (10)

The photon velocity *x* component is equal to *ET *plain velocity (2) (using (6)). Photon
*P* moves with the velocity (11) in *x* axis direction - measured in the units [U].

_{} (11)

We can calculate the photon orthogonal
velocity component in the *h* axis
direction (12) (measured in [U]) i.e. the photon *P* velocity in *ET *plain.

_{} (12)

We know all photon *P* velocity
components in moving co-ordinate system *k *(*w _{P}*

We can use previously described
transformation equations (4) and (5) (valid at *ET *plain)[a].

We gain (13) and (14) (both in [U]).

_{} (13)

_{} (14)

We can compute photon *P* total velocity
*w _{P}* in the stationary co-ordinate system

_{} (15)

We can evaluate (15) using (6) and gain (16).

_{} (16)

We can see directly from (16) (no measuring
unit) that photon *P* emitted from the moving light source is moving with
velocity grater than *c* in stationary co-ordinate system for any *b**>1* i.e. for any *v>0 *(and* t>0*). We can see from (16) that the photon *P
*velocity is dependent on the light source velocity *v*.

Figure 2 - Graphical representation of
the photon *P* total velocity *w _{P}* dependence on the light
source velocity

Described particular case shows the
existence of at least one situation where photon emitted from the moving light
source moves with velocity grater than *c* in the stationary co-ordinate
system and the photon velocity is dependent on the light source velocity. The
calculation is based on the correct use of SRT transformation including measuring
unit conversion. The result is in direct contradiction with the second
relativity theory postulate.

It must be noted, that the variant of the equal measuring units at all space axes can be taken into account. The geometrical distance inconsistency of SRT transformation described above exists in the particular case.

It means that at least one particular situation exists where SRT is internally inconsistent.

The particular case shows, that the proof
described in §3 ^{1} can not be generally valid. The existence of the
particular case proves that the second relativity theory postulate is
mathematically unsupported by SRT transformation.

[a] Equation (4) uses [u] and [U] units and (5) uses only [U] units. We must substitute velocity described in [u] units into (4) i.e. (10) to receive result in [U]. We must substitute velocity in [U] units into (5) i.e. (12) to receive results in [U].

[1] Albert Einstein 1905 On the Electrodynamics of Moving Bodies –
translated from „Zur Elektrodynamik betwegter Körper“ , Annalen der Physik,
17,1905 - *The Principle of Realitvity* – *A Collection of Original
Memories on the Special and General Theory of Relativity* – Mineola, N.Y.: Dover
Publication 1952, pages 35-65.