Master Special Relativity with Expert Help and Solutions

Why is it hopeless to view differential geometry as the limit of a discrete geometry?
Classical mechanics can be understood as the limit of relativistic mechanics $R{M}_c$ for $c\to <!--\; \to \; -->\mathrm{\infty <!--\; \infty \; -->}$.
Classical mechanics can be understood as the limit of quantum mechanics $Q{M}_h$ for $h\to <!--\; \to \; -->0$.
As a limit of which discrete geometry ${\mathrm{\Gamma <!--\; \Gamma \; -->}}_{\mathrm{\lambda <!--\; \lambda \; -->}}$ can classical mechanics be understood for $\mathrm{\lambda <!--\; \lambda \; -->}\to <!--\; \to \; -->0$?

If there were a light cone centered at some point P, and you were to look at that light cone from different reference frames, would it change its shape? I know that points inside and outside the light cone would remain inside/outside of the light cone in every frame, but does the light cone itself shift? If it does, how would it shift?

Imagine a space shuttle traveling through space at a constant velocity close to c. As the shuttle passes earth, a previously set-up camera starts broadcasting from earth to the shuttle. Since radio waves travel at the speed of light, the shuttle is receiving a constant transmission feed, assuming the camera is broadcasting 24/7.
Now, from what I have understood of special relativity so far, time will flow slower for the astronaut than for the earthlings. Hence, assuming $v=0.8c$, the astronaut will after 30 years have received a video transmission 50 years long!
Is my reasoning correct, that even though the transmission is live, the astronaut would actually be watching things that happened many years ago, while still receiving the "live" feed, which would be stored/buffered in the shuttles memory, thus making it possible for the astronaut to fast-forward the clip to see what happened more than 30 years after passing the earth?

A Newtonian homogeneous density sphere has gravitational binding energy in Joules $U=-<!--\; -\; -->(3/5)(G{M}^2)/r$, G=Newton's constant, M=gravitational mass, r=radius, mks. The fraction of binding energy to gravitational mass equivalent, $U/M{c}^2$, is then (-885.975 meters)(Ms/r), Ms = solar masses of body, c=lightspeed.
This gives ratios that are less than half that quoted for pulsars (neutron stars), presumably for density gradient surface to core and General Relativity effects (e.g., billion surface gees). Please post a more accurate formula acounting for the real world effects.
Examples: 1.74 solar-mass 465.1 Hz pulsar PSR J1903+0327, nominal radius 11,340 meters (AP4 model), calculates as 13.6% and is reported as 27%. A 2 sol neutron star calculates as 16.1% and is reported as 50%. There is an obvious nonlinearity.

Under the Lorentz transformations, quantities are classed as four-vectors, Lorentz scalars etc depending upon how their measurement in one coordinate system transforms as a measurement in another coordinate system.
The proper length and proper time measured in one coordinate system will be a calculated, but not measured, invariant for all other coordinate systems.
So what kind of invariants are proper time and proper length?

Why is travelling around the speed of light a problem? And why is it that the speed of light is the maximum?

Accelerating particles to speeds infinitesimally close to the speed of light?

While the speed of light in vacuum is a universal constant (c), the speed at which light propagates in other materials/mediums may be less than c. This is obviously suggested by the fact that different materials (especially in the case of transparent ones) have a particular refractive index.
But surely, matter or even photons can be accelerated beyond this speed in a medium?

Why the log? Is it there to make the growth of the function slower?
As this is a common experimental observable, it doesn't seem reasonable to take the range from $[0,\mathrm{\infty <!--\; \infty \; -->})$ to $(-<!--\; -\; -->\mathrm{\infty <!--\; \infty \; -->},\mathrm{\infty <!--\; \infty \; -->})$ (For a particle emitted along the beam axis after collision $\mathrm{\theta <!--\; \theta \; -->}=0$ wouldn't be better to have a number that says how close it is to zero rather than one that says how large a number it is. I hope that makes the question clear.)

If you have 2 flashlights, one facing North and one facing South, how fast are the photons (or lightbeams) from both flashlights moving away from one another?
Just adding speeds would yield 2C, but that's not possible as far as I know.
The reference frame here would be the place where the flashlights are and/or .The beams relative to one another.

If an observer were to rotate around a point at near light speeds, what sort of length contraction would he observe the universe undergo?