Learn Relativity Easily with Examples

From my understanding, we can do math in an accelerating frame of reference as long as "fictitious" force terms are correctly added. From this point of view, is there anything wrong with viewing the Earth as stationary, and the rest of the universe rotating around it, at least kinematically? And, if so, wouldn't several cosmological objects move faster than light in this frame of reference? How can this be?

If the velocity is a relative quantity, will it make inconsistent equations when applying it to the conservation of energy equations?
For example:
In the train moving at $V$ relative to ground, there is an object moving at $v$ relative to the frame in the same direction the frame moves. Observer on the ground calculates the object kinetic energy as $\frac{1}{2}m(v+V{)}^2$. However, another observer on the frame calculates the energy as $\frac{1}{2}m{v}^2$.

It is experimentally known that the equation of motion for a charge $e$ moving in a static electric field $\mathbf{E}$ is given by:
$\frac{\mathrm{d}}{\mathrm{d}t}(\mathrm{\gamma <!--\; \gamma \; -->}m\mathbf{v})=e\mathbf{E}$
Is it possible to show this using just Newton's laws of motion for the proper frame of $e$, symmetry arguments, the Lorentz transformations and other additional principles?

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?

The Fourier rate equation of heat conduction states:
$\stackrel{\to <!--\; \to \; -->}{q}=-<!--\; -\; -->k\mathrm{\nabla <!--\; \nabla \; -->}T$
But I'm wondering if this is valid in every frame of reference, because the heat flux $\stackrel{\to <!--\; \to \; -->}{q}$ does obviously change when the area under consideration is moving. The equation is often applied to moving fluids, and because there is no objective way to determine a non-moving frame of reference in a moving substance, I guess that the equation is valid for every inertial frame of reference, but I just wanted to be sure.

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.

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

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.

"You are traveling in a car going at a constant speed of 100 km/hr down a long, straight highway. You pass another car going in the same direction which is traveling at a constant speed of 80 km/hr. As measured from your car’s reference frame this other car is traveling at -20 km/hr. What is the acceleration of your car as measured from the other car’s reference frame? What is the acceleration of the other car as measured from your car’s reference frame?"

Shouldn't they both appear to have an acceleration of zero, because both velocities are constant? I can imagine sitting in the faster car and watching the slower car, its speed would not appear to change, only its position?

Let's take Faraday's experiments of induce emf where He take a loop and a magnetic field, for instance let's suppose we have an coil and a magnet, Now If we move the magnet towards the coil there will be an induce electric field and Thus the force $\stackrel{\to <!--\; \to \; -->}{F}=q\stackrel{\to <!--\; \to \; -->}{E}$ on charges of loop and thus the induce electric field but Now suppose I'm in frame of reference that is moving along with magnet with the same velocity as the magnet so that magnet is rest in my frame but coil is moving, in this case I will say charge in coil is moving in magnetic field and thus there is an Lorentz force $\stackrel{\to <!--\; \to \; -->}{F}=q(\stackrel{\to <!--\; \to \; -->}{v}\times <!--\; \times \; -->\stackrel{\to <!--\; \to \; -->}{B})$ and thus the induce emf. So How the same phenomenon is explained from two different Frame of reference with two different principals?

When I moving object in some frame of reference, it has a positive kinetic energy. In the frame of reference where the object is stationary, it has a kinetic energy of 0.
Is the energy difference in some way associated to the frames of reference?

Consider a person standing still on the ground and a car, which is moving with a constant speed $v$ in a straight line. From the frame of reference of the car the person is moving backwards with same speed $v$.
What causes to the person to move backwards in the frame of reference of the car?