In our daily experience, we live in a “non-relativistic” world with a very high degree of accuracy. Thus, if you see a train departing from you ( you are at rest relative to it) with speed
(in the positive direction of the x-axis), you move with relative velocity
respect to the train if you run in pursuit of it with speed
, or maybe you can also run with relative speed
if you run away from it in the opposite direction of motion.
However, light behavior is diferent to material bodies. Light, a.k.a. electromagnetic waves, is weird. I hope you have realized it from previous posts. We will see what happen with an SR analogue gedanken experiment of the previously mentioned “non-relativistic” train(S’-frame)-track(S-frame) experiment and that we have seen lot of times in our ordinary experience. We will discover that velocities close to the speed of light add in a different way, but we recover the classical result ( like the above) in the limit of low velocities ( or equivalently, in the limit
).
Problem to be solved:
In the S’-frame, an object (or particle) moves at constant velocity
relative to the S-frame. In the S’-frame, the object/particle moves with velocity
![Rendered by QuickLaTeX.com \[ \vec{u}\,'=\mathbf{u}'=(u'_x,u'_y,u'_z)=\left( \dfrac{dx'}{dt'},\dfrac{dy'}{dt'},\dfrac{dz'}{dt'}\right)\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-647d489691a730600accebf5153325d0_l3.png)
The question is: what is the velocity in the S-frame
![Rendered by QuickLaTeX.com \[ \vec{u}=\mathbf{u}=(u_x,u_y,u_z)=\left( \dfrac{dx}{dt},\dfrac{dy}{dt},\dfrac{dz}{dt}\right)\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-47badb8e81d6d3d798c955f3dedef585_l3.png)
CAUTION ( important note): v is constant (a relative velocity from one frame into another). u is not constant, since it is a vector describing the motion of a particle in some frame.
From the definiton of velocity, and the Lorentz transformation for a parallel motion, we have
![Rendered by QuickLaTeX.com \[ u_x=\dfrac{dx}{dt}=\dfrac{\dfrac{dx}{dt'}}{\dfrac{dt}{dt'}}=\dfrac{\dfrac{d}{dt'}\left[\gamma (x'+\beta ct')\right]}{\dfrac{d}{dt'}\left[\gamma (t'+\frac{\beta }{c}x')\right]}=\dfrac{\dfrac{dx'}{dt'}+\beta c \dfrac{dt'}{dt'}}{\dfrac{dt'}{dt'}+\dfrac{\beta}{c}\dfrac{dx'}{dt'}}\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-d18ef755c7318eca63f8e4b89733e945_l3.png)
and thus we get the addition law of velocities in the direction of motion
![Rendered by QuickLaTeX.com \[ \boxed{u_x=\dfrac{u'_x+v}{1+\dfrac{u'_x v}{c^2}}}\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-d85db96c34acfe122a005a42f6812130_l3.png)
We can also calculate the transformation of the transverse components to the velocity in the sense of motion. We only calculate the component
since the remaining one would be identical but labelled with other letter(the z-component indeed):
![Rendered by QuickLaTeX.com \[ u_y=\dfrac{dy}{dt}=\dfrac{\dfrac{dy}{dt'}}{\dfrac{dt}{dt'}}=\dfrac{\dfrac{dy'}{dt'}}{\gamma \left( 1+\dfrac{u'_x v}{c^2}\right)}\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-a4f0efbe458933f3143c1f78aabd8862_l3.png)
![Rendered by QuickLaTeX.com \[ \boxed{u_y=\dfrac{u'_y}{\gamma \left( 1+\dfrac{u'_x v}{c^2}\right)}}\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-c90dcd04b31cc74d62c4aa0b5b31122f_l3.png)
Therefore, the transverse velocity also changes in that way! There is an alternative deduction of the above formula using space and time coordinates. We will proceed in two important cases only.
The first case is when the motion happens with parallel relative velocity. Suppose two inertial frames S and S’. S is moving relative to S’ with velocity
along the X-axis. Moreover, suppose an object that is moving parallel to OX, with velocity
. Imagine two “frozen pictures” of the object at two different times according to S, e.g., fix two times
and
. The two events have coordinates of space and time given by
![Rendered by QuickLaTeX.com \[ E_1(t_1,x_1,y_1,z_1)\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-7fe2dd02762470a9ebccddce9035b142_l3.png)
and
![Rendered by QuickLaTeX.com \[ E_2(t_2,x_2,y_2,z_2)\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-c874dbfb4afdb68f75e71e8b896852be_l3.png)
But we do know that
and so,
![Rendered by QuickLaTeX.com \[ (t_2,x_2,y_2,z_2)=(t_1+\Delta t,x_1+v\Delta t,y_1,z_1)\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-51108e46f3d6748f1d6c2d7c6945b1b8_l3.png)
What does the S’-frame observe? Using the Lorentz boost with speed
, we get
![Rendered by QuickLaTeX.com \[ (t'_1,x'_1,y'_1,z'_1) = (\gamma (V)(t_1+Vx_1/c^2),\gamma (V) (x_1+Vt_1),y_1,z_1)\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-b30d4b0d6c4f8ad14207c1e2778a5698_l3.png)
and
![Rendered by QuickLaTeX.com \[ (t'_2,x'_2,y'_2,z'_2)=(\gamma (V)(t_2+Vx_2/c^2),\gamma (V)(x_2+Vt_2),y_2,z_2)\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-4a370696118e48e0bbfc81dc6d138b75_l3.png)
Therefore, the velocity of the object according to the S’-frame will be:
![Rendered by QuickLaTeX.com \[ v'=\dfrac{\Delta x'}{ \Delta t'}=\dfrac{x'_2-x'_1}{t'_2-t'_1}=\dfrac{x_2+Vt_2-(x_1+Vt_1)}{t_2+(V/c^2)x_2-(t_1+(V/c^2)x_1)}\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-d567f3bebb250f9e15f5d4c10b0cccf5_l3.png)
and in this way, we obtain, dividing by 
![Rendered by QuickLaTeX.com \[ v'=\dfrac{\dfrac{(x_2-x_1)+V(t_2-t_1)}{(t_2-t_1)}}{\dfrac{(t_2-t_1)+(V/c^2)(x_2-x_1)}{(t_2-t_1)}}\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-06b0befdfb4ab68ef10c285b8b7d2325_l3.png)
or equivalently
![Rendered by QuickLaTeX.com \[ \boxed{v'=\dfrac{v+V}{1+\dfrac{V \cdot v}{c^2}}}\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-d072945317e44b4c4151c4aa57c3d0ab_l3.png)
as before.
The second case is when in the velocities between the frames are orthogonal (perpendicular) can be also calculated in this way. Suppose that some object is moving in the orthogonal ( perpendicular) direction to the OX axis. For instance, we can suppose it moves along the y-axis (OY axis) with velocity v measured in the S-frame. We proceed in the same fashion that the previous calculation. We take two “imaginary pictures” of the body at
and
. We write the coordinates of space and time of this object as
![Rendered by QuickLaTeX.com \[ E_1(t_1,x_1,y_1,z_1)\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-7fe2dd02762470a9ebccddce9035b142_l3.png)
and
![Rendered by QuickLaTeX.com \[ E_2(t_2,x_2,y_2,z_2)\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-c874dbfb4afdb68f75e71e8b896852be_l3.png)
But we do know that
and so,
![Rendered by QuickLaTeX.com \[ (t_2,x_2,y_2,z_2)=(t_1+\Delta t,x_1,y_1+v\Delta t,z_1)\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-d4bd86a1a465ae5459c46934d24625d1_l3.png)
We make the corresponding Lorentz boost on those coordinates
![Rendered by QuickLaTeX.com \[ (t'_1,x'_1,y'_1,z'_1) = (\gamma (V)(t_1+Vx_1/c^2),\gamma (V) (x_1+Vt_1),y_1,z_1)\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-b30d4b0d6c4f8ad14207c1e2778a5698_l3.png)
![Rendered by QuickLaTeX.com \[ (t'_2,x'_2,y'_2,z'_2)=(\gamma (V)(t_2+Vx_2/c^2),\gamma (V)(x_2+Vt_2),y_2,z_2)\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-4a370696118e48e0bbfc81dc6d138b75_l3.png)
and now, the two components of this motion in the S’-frame will be given by ( note than our two set of coordinates have
and
in this particular case):
![Rendered by QuickLaTeX.com \[ v'_{x'}=\dfrac{\Delta x'}{\Delta t'}=\dfrac{x_2+Vt_2-(x_2+Vt_1)}{t_2+(V/c^2)x_2-(t_1+(V/c^2)x_1)}=V\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-126f975a15fd6ae2beb3daa67c46a2e7_l3.png)
![Rendered by QuickLaTeX.com \[ v'_{y'}=\dfrac{\Delta y'}{\Delta t'}=\dfrac{y'_2-y'_1}{\gamma (V)(t_1+(V/c^2)x_2-(t_1+(V/c^2)x_1))}=\dfrac{\dfrac{(y_2-y_1)}{(t_2-t_1)}}{\gamma (V)}=\dfrac{v}{\gamma (V)}\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-564e18a9ffbe7df0f827cb9129460e08_l3.png)
and so, in summary, in the orthogonal relative motion we have
![Rendered by QuickLaTeX.com \[ \boxed{ v'_{x'}=V \;\; v'_{y'}=\dfrac{v}{\gamma (V)}}\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-106ceb86f014909414a1432fde2a78c4_l3.png)
Indeed, these two cases are particular cases of the general transformations we got before.
The complete transformation of the velocity components and their inverses (obtained with the simple rule of mapping v into -v, and primed variables into unprimed variables) can be summarized by these formulae:
![Rendered by QuickLaTeX.com \[ \mbox{SR: Adding velocity(I)}\begin{cases}u_x=\dfrac{u'_x+v}{1+\dfrac{u'_x v}{c^2}} \; \; u_y=\dfrac{u'_y}{\gamma \left( 1+\dfrac{u'_x v}{c^2}\right)}\; \; u_z=\dfrac{u'_z}{\gamma \left( 1+\dfrac{u'_x v}{c^2}\right)}\\ \; \\ u'_x=\dfrac{u_x-v}{1-\dfrac{u_x v}{c^2}} \; \; u'_y=\dfrac{u_y}{\gamma \left( 1-\dfrac{u_x v}{c^2}\right)}\; \; u'_z=\dfrac{u_z}{\gamma \left( 1-\dfrac{u_x v}{c^2}\right)}\\ \; \\ \mathbf{u}=(u_x,u_y,u_z)\;\; \mathbf{u}'=(u'_x,u'_y,u'_z)\;\; \gamma =\dfrac{1}{\sqrt{1-\beta ^2}}\;\; \beta =\dfrac{v}{c}\end{cases}\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-6bf966450bdddc0b18bfb3861f9f103e_l3.png)
Of course, these transformations are valid in the case of a parallel relative motion between S and S’. What are the transformations in the case of non-parallel motion? Suppose that
![Rendered by QuickLaTeX.com \[ \vec{\beta} =\dfrac{\mathbf{v}}{c}=(\beta_x,\beta_y,\beta_z)\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-f313200e8db347ccefc5cf6d3cb069b6_l3.png)
and
as before. Then, using the most general Lorentz transformations
![Rendered by QuickLaTeX.com \[ \mathbf{u}'=\dfrac{d\mathbf{r}'}{dt'}=\dfrac{d\mathbf{r}+(\gamma - 1)\dfrac{\left(\vec{\beta}\cdot \mathbf{u}\right)\vec{\beta}}{\beta^2}-\gamma \vec{\beta}cdt}{\gamma dt - \dfrac{1}{c}\gamma \vec{\beta}\cdot d\mathbf{r}}\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-9b54a0c4047ea907970ba66515e4b4c6_l3.png)
then, using the same trick as above
![Rendered by QuickLaTeX.com \[ \mathbf{u}'=\dfrac{d\mathbf{r}'}{dt'}=\dfrac{\dfrac{d\mathbf{r}'}{dt}}{\dfrac{dt'}{dt}}\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-d4b235fa1e9596ed27b8fe9952e4cd3e_l3.png)
![Rendered by QuickLaTeX.com \[ \mathbf{u}'=\dfrac{\dfrac{1}{\gamma}\mathbf{u}+\left(1-\dfrac{1}{\gamma}\right)\dfrac{\left(\vec{\beta}\cdot \mathbf{u}\right)\vec{\beta}}{\beta^2}-\vec{\beta} c}{1-\dfrac{1}{c}\vec{\beta} \cdot \mathbf{u}}\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-ecbe31e4221c7f1a27233fdd1ecb2eea_l3.png)
We have got the following transformations (we apply the same recipe to obtain the inverse transformations, also included in the box below):
![Rendered by QuickLaTeX.com \[ \mbox{SR: Adding velocity(II)}\begin{cases} \mathbf{u}'=\dfrac{1}{1-\dfrac{\mathbf{u}\cdot\mathbf{v}}{c^2}}\left[\dfrac{\mathbf{u}}{\gamma}+\left[\left(1-\dfrac{1}{\gamma}\right)\dfrac{\mathbf{u}\cdot\mathbf{v}}{v^2}-1\right]\mathbf{v}\right]\\ \;\\ \mathbf{u}=\dfrac{1}{1+\dfrac{\mathbf{u}'\cdot\mathbf{v}}{c^2}}\left[\dfrac{\mathbf{u}'}{\gamma}-\left[-\left(1-\dfrac{1}{\gamma}\right)\dfrac{\mathbf{u}'\cdot\mathbf{v}}{v^2}-1\right]\mathbf{v}\right]\end{cases}\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-7f9343c365917e7b75e3205a5aff9c97_l3.png)
We observe that these equations are a non-linear addition of velocities. Equivalently, they can be rewritten as follows after some elementary algebra using a mathematical structure called gyrovector (or gyrovector addition):
![Rendered by QuickLaTeX.com \[ \mbox{gyrovector law}\begin{cases}\mathbf{u}'\equiv -\mathbf{v}\biguplus_{REL}\mathbf{u}=\dfrac{1}{1-\dfrac{\mathbf{u}\cdot \mathbf{v}}{c^2}}\left[-\mathbf{v}+\dfrac{\mathbf{u}}{\gamma_{\mathbf{v}}}+\dfrac{1}{c^2}\left(\dfrac{\gamma_{\mathbf{v}}}{\gamma_{\mathbf{v}}+1}\right)\left(\mathbf{v}\cdot \mathbf{u}\right)\mathbf{v}\right]\\ \; \\ \mathbf{u}\equiv \mathbf{v}\biguplus_{REL}\mathbf{u}'=\dfrac{1}{1+\dfrac{\mathbf{u}'\cdot \mathbf{v}}{c^2}}\left[\mathbf{v}+\dfrac{\mathbf{u}'}{\gamma_{\mathbf{v}}}+\dfrac{1}{c^2}\left(\dfrac{\gamma_{\mathbf{v}}}{\gamma_{\mathbf{v}}+1}\right)\left(\mathbf{v}\cdot \mathbf{u}'\right)\mathbf{v}\right]\end{cases}\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-7771883937a5430bd618551db1b933ed_l3.png)
These 2 cases can be seen as particular examples in the addition rule of velocities as a “gyrovector sum”, the nonlinear addition rule given by the formula:
![Rendered by QuickLaTeX.com \[ \mathbf{u}\biguplus_{REL}\mathbf{v}=\dfrac{1}{1+\dfrac{\mathbf{u}\cdot \mathbf{v}}{c^2}}\left[\mathbf{u}+\dfrac{\mathbf{v}}{\gamma_{\mathbf{u}}}+\dfrac{1}{c^2}\left(\dfrac{\gamma_{\mathbf{u}}}{\gamma_{\mathbf{u}}+1}\right)\left(\mathbf{u}\cdot \mathbf{v}\right)\mathbf{u}\right]\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-7b6baf56f2df81569e2e0b7270a41d45_l3.png)
This formula is usually written in a more intuitive expression with the following arguments. Suppose some object moves with velocity
in some inertial frame S. S is moving itself with relative velocity
respect to another frame S’. In the S’-frame, the velocity is given by:
![Rendered by QuickLaTeX.com \[ \boxed{\mathbf{v}'\equiv \mathbf{V}\biguplus_{REL}\mathbf{v}=\dfrac{\mathbf{v}_\parallel+\gamma^{-1}(V)\mathbf{v}_\perp + \mathbf{V}}{1+\dfrac{\mathbf{V}\cdot \mathbf{v}}{c^2}}}\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-8984bca360ee2bbd59b4895274c8ab1d_l3.png)
and where we have defined the projections of
in the direction parallel and orthogonal to
. They are given by:
![Rendered by QuickLaTeX.com \[ \boxed{\mathbf{v}_\parallel =\dfrac{(\mathbf{V}\cdot \mathbf{u})\mathbf{V}}{V^2}\;\;\;\;\;\; V^2=\vert \mathbf{V}\vert^2\;\;\;\;\;\mathbf{v}_\perp =\mathbf{v}-\mathbf{v}_\parallel}\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-d2264a1d20ad4315cd8103de8f905618_l3.png)
We will talk about gyrovectors more in a future post. They have a curious mathematical structure and geometry, and they are not well known by physicists since they are not in the basic curriculum and background of SR courses. Of course, the non-associative composition rule for velocities is not a standard formula you can find in books about relativity, so I will write it here:
![Rendered by QuickLaTeX.com \[ \boxed{\mathbf{u}\boxplus\mathbf{v}=\dfrac{\mathbf{u}+\mathbf{v}}{1+\dfrac{\mathbf{u}\cdot \mathbf{v}}{c^2}}+\dfrac{\gamma_{\mathbf{u}}}{c^2(\gamma_{\mathbf{u}}+1)}\dfrac{\mathbf{u}\times\left(\mathbf{u}\times\mathbf{v}\right)}{1+\dfrac{\mathbf{u}\cdot \mathbf{v}}{c^2}}}\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-4cf67052406c1daac0161062b2b48d63_l3.png)
and where we used the previous formula for
and after some algebra we used the known relationship for the cross product of three vectors, two being the same,
![Rendered by QuickLaTeX.com \[ u\times(u\times v)=(u\cdot v)u-(u\cdot u)v\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-8105b7cabfccc02cfc3f64484522477c_l3.png)
Let’s go back to the beginning. Suppose now we imagine a train (our S’-frame) travelling at velocity
. Suppose that Special Relativity matters now. We are in the tracks as observers “in relative rest” with respect to the train ( we are the S-frame) and suppose that we take into account the SR corrections above to the addition of velocities. Inside the train some object is being thrown with velocity
in the direction of motion. What is the velocity
in the S-frame? That is, what is the velocity we observe in the tracks? In this simple example, we use the easier addition rule of velocities ( named addition rule SR(I) above). Firstly, we note some expected features from the mathematical structure of the relativistic addition rule of velocities (valid propterties as well in the general case (II) with a suitable generalization):
1st. For low velocities, i.e., if
and/or
, the result approaches the nonrelativistic “ordinary life” experience:
.
2nd. For positive velocities
and a positive relative velocity between frames
, the addition of velocities is generally
, i.e., we get a velocity smaller that in the non-relativistic (ordinary or “common” experience) limit.
Now, some easy numerical examples to see what is going on bewteen the train (S’) and the track (S) where we are:
Example 1. Moderate velocity case. We have, e.g., velocities
. This gives, using (I):

Then, the deviation with respect to the non-relativistic value ( 60km/s) is negligible for all the practical purposes! This typical velocity, 30km/s, is about the typical velocities in 20th and early 21st century space flight. So, our astronauts can not note/observe relativistic effects. The addition theorem in SR is not practical in current space travel (20th/early 21st century).
Example 2. Case velocities are “close enough” to the speed of light. E.g.: One quarter and one half of the speed of light. In the first case,
![Rendered by QuickLaTeX.com \[ v=u'_x=0.25c\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-4338cd244a1d9a6b17d1ba851e8d568b_l3.png)
and then
![Rendered by QuickLaTeX.com \[ u_x=\dfrac{0.25c+0.25c}{1+0.0625}=\dfrac{0.5c}{1.0625}\approx 0.47c\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-eb624c0f09d0ecd0b91c1c567b7e62a5_l3.png)
In the seconde case, we write
. This provides
![Rendered by QuickLaTeX.com \[ u_x=\dfrac{0.5c+0.5c}{1+0.25}=\dfrac{c}{1.25}=0.8c\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-b6deb3dbd0254c3e56389555d3cee59a_l3.png)
Thus, we observe that a higher velocities, e.g., those in particle accelerators, some processes in the Universe, etc, the relativistic effects of the non-linear addition of velocities can NOT be neglected. The effect is important and becomes increasingly important when the velocity increases itself ( you can note how large the SR effect is if you compare the 0.25c and 0.5c examples above).
Example 3. Speed of light case. Extreme case: we are trying to exceed the velocity of light. Suppose now, that the train could move with relative velocity equal to c. The object is thrown with relative speed
and
. What we do see on the track. Naively, ordinary life would suggest the answer 2c, but we do know that velocities transform non-linearly, so, we plug the values in the formula to get the answer:
and
. Therefore, if a train is travelling at the speed of light, and inside the train an object is thrown forward at c, we DO NOT observe/measure a 2c velocity, we observe/measure it has velocity
!!!! if we stand at rest on the track. Amazing!
Suppose we try to do it in a “transverse way”. That is, suppose that the velocities are now
,
and the transverse speed is not
. This case results in the numbers:
![Rendered by QuickLaTeX.com \[ u_x=c\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-ec4c1daa11569c893f6da76773cf2b86_l3.png)
and
![Rendered by QuickLaTeX.com \[ u_y=\dfrac{c}{\gamma}=0\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-cfd2f172709f6e851f6eb0a48f4f2bae_l3.png)
since
![Rendered by QuickLaTeX.com \[ \gamma \rightarrow \infty\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-8392f135655bdfae7746feb1ff461d4a_l3.png)
and thus
.
Therefore, if the train travels at c, an inside of the train an object is launched at right angles to the direction of motion at the velocity c, the object itself is measured/seen to have velocity c measured from a rest observer placed beside the tracks. Amazing, surprise again!
Example 4. Case: Superluminal relative motion. Suppose, somehow, the relative motion between the two frames provides
(even you can plug
with
if you wish). Suppose the object is measured to have the extremal limit speed
(imagine we consider a light beam/flash, for instance). Again, using the addition law we would get:
![Rendered by QuickLaTeX.com \[ u_x=\dfrac{3c}{1+2}=c\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-7750994a91a430093a45a37e86cc7949_l3.png)
and
![Rendered by QuickLaTeX.com \[ u_x=\dfrac{n+1}{1+n}c=c\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-29f6f1d6a8d98b9348ee8a02821a0389_l3.png)
Even if the inertial frames move at superluminal velocities relative to each other, a light beam would remain c in the S-frame if SR holds! Surprise, again!
In this way, we can conclude one of the most important conclusions of special relativity ( something that it is ignored by many Sci-fi writers, and that we would like to be able to overcome somehow if we have to master the interstellar travel/interstellar communications as Sci-fi fans, or as an interstellar civilization, you should get some trick to avoid/”live with” it.):
![Rendered by QuickLaTeX.com \[ \boxed{\mbox{The speed of light can never be exceeded by adding velocities in SR.}}\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-a2748c2ab12c90ca8ce0f19221b2891e_l3.png)
If SR holds, the velocity (or speed) of light is the maximum speed attainable in the Universe. You can like it or hate it, but if SR is true, you can not avoid this conclusion.
There is another special case of motion important in practical applications: two dimensional motion. I mean, imagine that in the S’-frame, an object has the velocity
. The velocity subtends an angle
with the x’-axis. See the figure below:

What is the angle
in the frame that we observe between
and the x-axis? For the S-frame we find:
![Rendered by QuickLaTeX.com \[\tan \theta= \dfrac{u_y}{u_x}=\dfrac{\dfrac{u'_y}{\gamma \left( 1+\dfrac{u'_x v}{c^2}\right)}}{\dfrac{u'_x+v}{1+\dfrac{u'_x v}{c^2}}}\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-9a510b930035757d2c4fcb34c674899d_l3.png)
and after some easy algebraic manipulations we get the important result
![Rendered by QuickLaTeX.com \[ \boxed{\tan \theta =\dfrac{1}{\gamma}\dfrac{u'_y}{u'_x+v}}\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-3725f3b07c904ab8a4067638a2c2a34d_l3.png)
We observe that, according to this last equation, with teh exception of
,
is smaller in the S frame than
in the S’-frame. In the non-relativist limit, we recover the result that our ordinary intuition and experience provides (
):
![Rendered by QuickLaTeX.com \[ \theta_{nonrel}=\dfrac{u'_y}{u'_x+v}\]](http://www.thespectrumofriemannium.com/wp-content/ql-cache/quicklatex.com-f60a44f91b7a404f7298a8fb79dfd39c_l3.png)
It is logical. In the nonrelativistic limit we do know that
and
, so the result agrees with our experience in the low velocity realm.
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