Master Special Relativity with Expert Help and Solutions

It would be nice to have a cute method that uses Lorentz transformations of basis vectors by exponential transformation using gamma matrices. To avoid confusion, let's assume $-<!--\; -\; -->+++$ signature. Given ${\mathrm{\gamma <!--\; \gamma \; -->}}_{\mathrm{\mu <!--\; \mu \; -->}}$ as gamma matrices that satisfy $\mathrm{t}\mathrm{r}({\mathrm{\gamma <!--\; \gamma \; -->}}_{\mathrm{\mu <!--\; \mu \; -->}}{\mathrm{\gamma <!--\; \gamma \; -->}}_{\mathrm{\nu <!--\; \nu \; -->}})=4{\mathrm{\eta <!--\; \eta \; -->}}_{\mathrm{\mu <!--\; \mu \; -->}\mathrm{\nu <!--\; \nu \; -->}}$, then we have $\mathrm{t}\mathrm{r}({\mathrm{\gamma <!--\; \gamma \; -->}}_{\mathrm{\mu <!--\; \mu \; -->}}^{\prime }{\mathrm{\gamma <!--\; \gamma \; -->}}_{\mathrm{\nu <!--\; \nu \; -->}}^{\prime })=4{\mathrm{\eta <!--\; \eta \; -->}}_{\mathrm{\mu <!--\; \mu \; -->}\mathrm{\nu <!--\; \nu \; -->}}$ if we put:${\mathrm{\gamma <!--\; \gamma \; -->}}_{\mathrm{\mu <!--\; \mu \; -->}}^{\prime }=\exp <!--\; \; -->(-<!--\; -\; -->A){\mathrm{\gamma <!--\; \gamma \; -->}}_{\mathrm{\mu <!--\; \mu \; -->}}\exp <!--\; \; -->(+A)$where $A$ is any matrix.
Boosts and rotations use $A$ as a bivector. For example, with $\mathrm{\alpha <!--\; \alpha \; -->}$ a real number, $A=\mathrm{\alpha <!--\; \alpha \; -->}{\mathrm{\gamma <!--\; \gamma \; -->}}_0{\mathrm{\gamma <!--\; \gamma \; -->}}_3$ boosts in the $z$ direction while $A=\mathrm{\alpha <!--\; \alpha \; -->}{\mathrm{\gamma <!--\; \gamma \; -->}}_1{\mathrm{\gamma <!--\; \gamma \; -->}}_2$ gives a rotation around the $z$ axis.
Solving for the value of α that gives a boost with velocity $\mathrm{\beta <!--\; \beta \; -->}=v/c$ appears to be straightforward. But how to do the rotations?
And by the way, what happens when you generalize $A$ to be something other than bivectors?

In history we are taught that the Catholic Church was wrong, because the Sun does not move around the Earth, instead the Earth moves around the Sun.
But then in physics we learn that movement is relative, and it depends on the reference point that we choose.
Wouldn't the Sun (and the whole universe) move around the Earth if I place my reference point on Earth?
Was movement considered absolute in physics back then?

Looking for specific Relativity example
The example had to do with two people walking along a sidewalk in opposite directions, and an alien race on a planet millions of light-years away planning an invasion of the Solar System. The example showed that in one walker's reference frame the invasion fleet had departed, but in the other reference frame the fleet had not.
At the time, the explanation made perfect sense, but I have forgotten the details and have never run across this example again.
Does anybody know where this was, or have the text of the explanation?

Suppose we have a sphere of radius $r$ and mass m and a negatively charged test particle at distance d from its center, $d\gg <!--\; \gg \; -->r$. If the sphere is electrically neutral, the particle will fall toward the sphere because of gravity. As we deposit electrons on the surface of the sphere, the Coulomb force will overcome gravity and the test particle will start to accelerate away. Now suppose we keep adding even more electrons to the sphere. If we have n electrons, the distribution of their pairwise distances has a mean proportional to $r$, and there are $n(n-<!--\; -\; -->1)/2$ such pairs, so the binding energy is about $n^2/r$. If this term is included in the total mass-energy of the sphere, the gravitational force on the test particle would seem to be increasing quadratically with $n$, and therefore eventually overcomes the linearly-increasing Coulomb force. The particle slows down, turns around, and starts falling again. This seems absurd; what is wrong with this analysis?

Does a relativistic version of quantum thermodynamics exist? I.e. in a non-inertial frame of reference, can I, an external observer, calculate quantities like magnetisation within the non-inertial frame?

Einstein said that the synchronization of two clocks is dependant on the velocity of the observer. But I feel a conceptual contradiction can be made:
There are two observers A and B. Observer 'A' faces direction X, and will be labeled "stationary." Another observer B faces direction X and is travelling rapidly in that direction as well on a collision course with 'A'. Both observers are holding two clocks; One in each hand, with hands held perpendicular to direction X. Both observers hit a "synchronization" button on the clocks before colliding.
My expectation in this case is that when the moving observer B halts to greet observer A - both observers will agree the B-pair clocks are synchronized and the A-pair clocks are synchronized (though all four clocks are not necessarily synchronized with eachother, and is not at issue in this question).
Bottom line: what was considered "synchronicity" by the speedy individual is accepted by the stationary individual. This seems to contradict special relativity.
Now if my scenario were modified slightly, then I believe special relativity would apply. If observer A were to hold the A-pair clocks with one held out before herself and the other held out behind herself (instead of the original left and right perpendicular angles), then finally I would expect there to be disagreement between the two observers when they stop to meet eachother.
All this suggests that position is an unaccounted for primary component of relativity, but from what little I know of SR, it doesn't hardly factor in at all. Can someone explain what I'm missing?

Is there a time + two spatial dimension representation of a Minkowski-space surface which could be constructed within our own (assumed Euclidean) 3D space such that geometric movement within the surface would intuitively demonstrate the “strange" effects of the Lorentz transformation (length contraction, time dilation)?

It seems odd that entropy is usually only defined for a system in a single 'slice' of time or spacelike region. Can one define the entropy of a system defined by a 4d region of spacetime, in such a way that yields a codimension one definition which agrees with the usual one when the codimension one slice is spacelike?

I know that as one approaches the speed of light, time moves slower for him. So, if I start moving as fast as 99% of the speed of light and travel away from Earth for 1 day and come back, I'll see that (suppose) about 1 year has passed on Earth.
Question is, if everything is relative, then how can we say that I was the one moving and Earth was the one staying?

An electron is shot towards a target that is negatively charged. While the electron is traveling, the target makes an abrupt move towards the electron. While the information that the target moved is traveling from the target to the electron, the electron behaves like an electron that is moving towards a target that is in the original position.
How can energy be conserved when an electron that is moving towards a nearby charge behaves like it was moving towards a far away charge? Seems we end up with electron being at 2 meters distance from the target, while the electron had enough energy to travel to at most 4 meters distance from the target.
It also seems to me that "moving the target requires energy" is not a solution to this problem.

It is well known that quantum mechanics and (general) relativity do not fit well. I am wondering whether it is possible to make a list of contradictions or problems between them?
E.g. relativity theory uses a space-time continuum, while quantum theory uses discrete states.

Environmentally induced decoherence makes wave function collapse unnecessary. But the environment, usually taken to be some heat bath, introduces a preferred frame. (That in which the total (spatial) momentum vanishes.) So, doesn't then the decoherence time depend on the motion of the prepared state relative to the environment? And, doesn't the ultimate environment, all particles in the universe, introduce a preferred frame into quantum mechanics in the sense that the decoherence time is relative to this frame? And would this be measureable, at least in principle

I am asked to show that the matrix
$S=1-<!--\; -\; -->\frac{i}{4}{\mathrm{\sigma <!--\; \sigma \; -->}}_{\mathrm{\mu <!--\; \mu \; -->}\mathrm{\nu <!--\; \nu \; -->}}{\mathrm{\epsilon <!--\; \epsilon \; -->}}^{\mathrm{\mu <!--\; \mu \; -->}\mathrm{\nu <!--\; \nu \; -->}}$
represents the infinitesimal Lorentz transformation
${\mathrm{\Lambda <!--\; \Lambda \; -->}}^{\mathrm{\mu <!--\; \mu \; -->}}{}_{\mathrm{\nu <!--\; \nu \; -->}}={\mathrm{\delta <!--\; \delta \; -->}}^{\mathrm{\mu <!--\; \mu \; -->}}{}_{\mathrm{\nu <!--\; \nu \; -->}}+{\mathrm{\epsilon <!--\; \epsilon \; -->}}^{\mathrm{\mu <!--\; \mu \; -->}}{}_{\mathrm{\nu <!--\; \nu \; -->}},$
in the sense that
$S^{-<!--\; -\; -->1}{\mathrm{\gamma <!--\; \gamma \; -->}}^{\mathrm{\mu <!--\; \mu \; -->}}S={\mathrm{\Lambda <!--\; \Lambda \; -->}}^{\mathrm{\mu <!--\; \mu \; -->}}{}_{\mathrm{\nu <!--\; \nu \; -->}}{\mathrm{\gamma <!--\; \gamma \; -->}}^{\mathrm{\nu <!--\; \nu \; -->}}.$
I have already proven that $S^{-<!--\; -\; -->1}={\mathrm{\gamma <!--\; \gamma \; -->}}^0{S}^{†<!--\; †\; -->}{\mathrm{\gamma <!--\; \gamma \; -->}}^0$, so I can begin with the left-hand side of this third equation:
$\begin{array}{rl}{S}^{-<!--\; -\; -->1}{\mathrm{\gamma <!--\; \gamma \; -->}}^{\mathrm{\mu <!--\; \mu \; -->}}S& ={\mathrm{\gamma <!--\; \gamma \; -->}}^0{S}^{†<!--\; †\; -->}{\mathrm{\gamma <!--\; \gamma \; -->}}^0{\mathrm{\gamma <!--\; \gamma \; -->}}^{\mathrm{\mu <!--\; \mu \; -->}}S\\ & ={\mathrm{\gamma <!--\; \gamma \; -->}}^0(1+\frac{i}{4}{\mathrm{\sigma <!--\; \sigma \; -->}}_{\mathrm{\mu <!--\; \mu \; -->}\mathrm{\nu <!--\; \nu \; -->}}{\mathrm{\epsilon <!--\; \epsilon \; -->}}^{\mathrm{\mu <!--\; \mu \; -->}\mathrm{\nu <!--\; \nu \; -->}}){\mathrm{\gamma <!--\; \gamma \; -->}}^0{\mathrm{\gamma <!--\; \gamma \; -->}}^{\mathrm{\mu <!--\; \mu \; -->}}(1-<!--\; -\; -->\frac{i}{4}{\mathrm{\sigma <!--\; \sigma \; -->}}_{\mathrm{\mu <!--\; \mu \; -->}\mathrm{\nu <!--\; \nu \; -->}}{\mathrm{\epsilon <!--\; \epsilon \; -->}}^{\mathrm{\mu <!--\; \mu \; -->}\mathrm{\nu <!--\; \nu \; -->}})\\ & ={\mathrm{\gamma <!--\; \gamma \; -->}}^0{\mathrm{\gamma <!--\; \gamma \; -->}}^0{\mathrm{\gamma <!--\; \gamma \; -->}}^{\mathrm{\mu <!--\; \mu \; -->}}-<!--\; -\; -->{\mathrm{\gamma <!--\; \gamma \; -->}}^0{\mathrm{\gamma <!--\; \gamma \; -->}}^0{\mathrm{\gamma <!--\; \gamma \; -->}}^{\mathrm{\mu <!--\; \mu \; -->}}\frac{i}{4}{\mathrm{\sigma <!--\; \sigma \; -->}}_{\mathrm{\mu <!--\; \mu \; -->}\mathrm{\nu <!--\; \nu \; -->}}{\mathrm{\epsilon <!--\; \epsilon \; -->}}^{\mathrm{\mu <!--\; \mu \; -->}\mathrm{\nu <!--\; \nu \; -->}}+{\mathrm{\gamma <!--\; \gamma \; -->}}^0\frac{i}{4}{\mathrm{\sigma <!--\; \sigma \; -->}}_{\mathrm{\mu <!--\; \mu \; -->}\mathrm{\nu <!--\; \nu \; -->}}{\mathrm{\epsilon <!--\; \epsilon \; -->}}^{\mathrm{\mu <!--\; \mu \; -->}\mathrm{\nu <!--\; \nu \; -->}}{\mathrm{\gamma <!--\; \gamma \; -->}}^0{\mathrm{\gamma <!--\; \gamma \; -->}}^{\mathrm{\mu <!--\; \mu \; -->}}-<!--\; -\; -->{\mathrm{\gamma <!--\; \gamma \; -->}}^0\frac{i}{4}{\mathrm{\sigma <!--\; \sigma \; -->}}_{\mathrm{\rho <!--\; \rho \; -->}\mathrm{\tau <!--\; \tau \; -->}}{\mathrm{\epsilon <!--\; \epsilon \; -->}}^{\mathrm{\rho <!--\; \rho \; -->}\mathrm{\tau <!--\; \tau \; -->}}{\mathrm{\gamma <!--\; \gamma \; -->}}^0{\mathrm{\gamma <!--\; \gamma \; -->}}^{\mathrm{\mu <!--\; \mu \; -->}}\frac{i}{4}{\mathrm{\sigma <!--\; \sigma \; -->}}_{\mathrm{\mu <!--\; \mu \; -->}\mathrm{\nu <!--\; \nu \; -->}}{\mathrm{\epsilon <!--\; \epsilon \; -->}}^{\mathrm{\mu <!--\; \mu \; -->}\mathrm{\nu <!--\; \nu \; -->}}\\ & ={\mathrm{\gamma <!--\; \gamma \; -->}}^{\mathrm{\mu <!--\; \mu \; -->}}+\frac{1}{16}{\mathrm{\gamma <!--\; \gamma \; -->}}^0{\mathrm{\sigma <!--\; \sigma \; -->}}_{\mathrm{\rho <!--\; \rho \; -->}\mathrm{\tau <!--\; \tau \; -->}}{\mathrm{\epsilon <!--\; \epsilon \; -->}}^{\mathrm{\rho <!--\; \rho \; -->}\mathrm{\tau <!--\; \tau \; -->}}{\mathrm{\gamma <!--\; \gamma \; -->}}^0{\mathrm{\gamma <!--\; \gamma \; -->}}^{\mathrm{\mu <!--\; \mu \; -->}}{\mathrm{\sigma <!--\; \sigma \; -->}}_{\mathrm{\mu <!--\; \mu \; -->}\mathrm{\nu <!--\; \nu \; -->}}{\mathrm{\epsilon <!--\; \epsilon \; -->}}^{\mathrm{\mu <!--\; \mu \; -->}\mathrm{\nu <!--\; \nu \; -->}}\\ & ={\mathrm{\gamma <!--\; \gamma \; -->}}^{\mathrm{\mu <!--\; \mu \; -->}}+\frac{1}{16}\left({\mathrm{\sigma <!--\; \sigma \; -->}}_{\mathrm{\rho <!--\; \rho \; -->}\mathrm{\tau <!--\; \tau \; -->}}{\mathrm{\epsilon <!--\; \epsilon \; -->}}^{\mathrm{\rho <!--\; \rho \; -->}\mathrm{\tau <!--\; \tau \; -->}}{\mathrm{\sigma <!--\; \sigma \; -->}}_{\mathrm{\mu <!--\; \mu \; -->}\mathrm{\nu <!--\; \nu \; -->}}{\mathrm{\epsilon <!--\; \epsilon \; -->}}^{\mathrm{\mu <!--\; \mu \; -->}\mathrm{\nu <!--\; \nu \; -->}}\right){\mathrm{\gamma <!--\; \gamma \; -->}}^{\mathrm{\mu <!--\; \mu \; -->}}.\end{array}$
Remember, we want this to be equal to
${\mathrm{\Lambda <!--\; \Lambda \; -->}}^{\mathrm{\mu <!--\; \mu \; -->}}{}_{\mathrm{\nu <!--\; \nu \; -->}}{\mathrm{\gamma <!--\; \gamma \; -->}}^{\mathrm{\nu <!--\; \nu \; -->}}=({\mathrm{\delta <!--\; \delta \; -->}}^{\mathrm{\mu <!--\; \mu \; -->}}{}_{\mathrm{\nu <!--\; \nu \; -->}}+{\mathrm{\epsilon <!--\; \epsilon \; -->}}^{\mathrm{\mu <!--\; \mu \; -->}}{}_{\mathrm{\nu <!--\; \nu \; -->}}){\mathrm{\gamma <!--\; \gamma \; -->}}^{\mathrm{\nu <!--\; \nu \; -->}}={\mathrm{\gamma <!--\; \gamma \; -->}}^{\mathrm{\mu <!--\; \mu \; -->}}+{\mathrm{\epsilon <!--\; \epsilon \; -->}}^{\mathrm{\mu <!--\; \mu \; -->}}{}_{\mathrm{\nu <!--\; \nu \; -->}}{\mathrm{\gamma <!--\; \gamma \; -->}}^{\mathrm{\nu <!--\; \nu \; -->}}.$
The first term is there already, but I have no idea how that second term is going to work out to ${\mathrm{\epsilon <!--\; \epsilon \; -->}}^{\mathrm{\mu <!--\; \mu \; -->}}{}_{\mathrm{\nu <!--\; \nu \; -->}}{\mathrm{\gamma <!--\; \gamma \; -->}}^{\mathrm{\nu <!--\; \nu \; -->}}$. Can anyone give me a hint, or tell me what I’ve done wrong?

if $u$ and $u^{\prime }$ are a velocity referred to two inertial frames with relative velocity $v$ confined to the $x$ axis, then the quantities $l$, $m$, $n$ defined by
$(l,m,n)=\frac{1}{|u|}(u_x,u_y,u_z)$
and
$(l^{\prime },m^{\prime },n^{\prime })=\frac{1}{|u^{\prime }|}(u_x^{\prime },u_y^{\prime },u_z^{\prime })$
are related by
$(l^{\prime },m^{\prime },n^{\prime })=\frac{1}{D}(l-<!--\; -\; -->\frac{v}{u},m{\mathrm{\gamma <!--\; \gamma \; -->}}^{-<!--\; -\; -->1},n{\mathrm{\gamma <!--\; \gamma \; -->}}^{-<!--\; -\; -->1})$
and that this can be considered a relativistic aberration formula. The author gives the following definition for $D$, copied verbatim.
$D=\frac{u^{\prime }}{u}(1-<!--\; -\; -->\frac{u_{x}v}{c^2})={[1-<!--\; -\; -->2l\frac{v}{u}+\frac{v^2}{u^2}-<!--\; -\; -->(1-<!--\; -\; -->l^2)\frac{v^2}{c^2}]}^{\frac{1}{2}}$
Why is that better than the second expression?
Also, in case it's not clear, $\mathrm{\gamma <!--\; \gamma \; -->}=1/\sqrt{1-<!--\; -\; -->\frac{v^2}{c^2}}$ and $|u|=|(u_x,u_y,u_z)|=\sqrt{u_x^2+u_y^2+u_z^2}$

Chosing a reference frame in which the Earth is at rest and doesn't rotate
1) Does anybody know of such a publication?
2) I know that even such speeds of ${10}^{18}$ m/s are not in contradiction with relativity because a limiting velocity only exists for exchange of information, which apparantly does not occur.

The relativistic energy-momentum equation is:
$E^2=(pc{)}^2+(m{c}^2{)}^2.$
Also, we have $pc=Ev/c$, so we get:
$E=m{c}^2/(1-<!--\; -\; -->v^2/c^2{)}^{1/2}.$
Now, accelerating a proton to near the speed of light:
$\begin{array}{}0.990000000000000& c=>& 0.0000000011& J=& 0.01& TeV\\ 0.999000000000000& c=>& 0.0000000034& J=& 0.02& TeV\\ 0.999900000000000& c=>& 0.0000000106& J=& 0.07& TeV\\ 0.999990000000000& c=>& 0.0000000336& J=& 0.21& TeV\\ 0.999999000000000& c=>& 0.0000001063& J=& 0.66& TeV\\ 0.999999900000000& c=>& 0.0000003361& J=& 2.10& TeV\\ 0.999999990000000& c=>& 0.0000010630& J=& 6.64& TeV\\ 0.999999999000000& c=>& 0.0000033614& J=& 20.98& TeV\\ 0.999999999900000& c=>& 0.0000106298& J=& 66.35& TeV\\ 0.999999999990000& c=>& 0.0000336143& J=& 209.83& TeV\\ 0.999999999999000& c=>& 0.0001062989& J=& 663.54& TeV\\ 0.999999999999900& c=>& 0.0003360908& J=& 2,097.94& TeV\\ 0.999999999999990& c=>& 0.0010634026& J=& 6,637.97& TeV\\ 0.999999999999999& c=>& 0.0033627744& J=& 20,991.10& TeV\end{array}$
If the LHC is accelerating protons to $7TeV$ it means they're traveling with a speed of $0.99999999c$.
Is everything above correct?

instead of assuming that the velocity $c$ is a maximal velocity, proving that while assuming $E=m{c}^2$.

If the light velocity is a vector quantity, why vector addition cannot be applied to it? Or the light velocity is not a vector quantity?

Take the following gedankenexperiment in which two astronauts meet each other again and again in a perfectly symmetrical setting - a hyperspherical (3-manifold) universe in which the 3 dimensions are curved into the 4. dimension so that they can travel without acceleration in straight opposite directions and yet meet each other time after time.
On the one hand this situation is perfectly symmetrical - even in terms of homotopy and winding number. On the other hand the Lorentz invariance should break down according to GRT, so that one frame is preferred - but which one?
So the question is: Who will be older? And why?
And even if there is one prefered inertial frame - the frame of the other astronaut should be identical with respect to all relevant parameters so that both get older at the same rate. Which again seems to be a violation of SRT in which the other twin seems to be getting older faster/slower...
How should one find out what the preferred frame is when everything is symmetrical - even in terms of GRT

Observer A and B are at the same "depth" in a gravity well. Observer B then descends into the well. A will observe B's time as going slower than their own. B will observe A's time as going faster than their own.
What happens if B were to ascend the well back to A's depth, would B's local time speed back up to the same rate as A's, but B would be younger (relative to A)?
What about the paradox caused by relative motion (ignoring gravity)? If A is moving relative to B, A and B will both observe the other's time as going slower. If A and B were together initially, then B moves away and returns, do their clocks agree?