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From what height would we need to „drop“ an object on a neutron star to make it land with a speed 0,1 $c$ (0,1 of speed of light). Our neutron star is 1.5 times heavier than our Sun and has diameter $d=10\;\mathrm{km}$. Ignore both the atmosphere of the star and its rotation. You can also ignore the correction for special relativity. However, do compare the results for a homogenous gravitational field (with the same strength as is on the star surface) and for a radial gravitational field. Bonus: Do not ignore the special relativity correction.

Superman and Flash decided to race each other. The race takes place in deep space as there is no straight beach long enough on Earth. As Flash is slower, he starts with a length lead $l$ ahead of Superman. At one moment, Flash starts with a constant speed $v_{F}$ comparable with the speed of light. At the moment Superman sees that Flash started, he starts running at a constant speed $v_{S}>v_{F}$. How long will it take Superman to catch up with Flash (from Superman's point of view)? How long will it take from Flash's point of view? Was the starting method fair? Can you devise a more fair method (keeping the length lead $l)?$

Lukas has been weightlifting and he managed to make a black hole of mass 1 kg. As he isn't too fond of quantum field theory in curved spacetime, the black hole does not radiate. Lukas drops this hole and it begins oscillating within the earth. Try to estimate how long would it take for the mass of the black hole to double. Is it safe to make black holes at home?

During his visit to Ankh-Morpork Two flower also visited a pub. It wouldn't have been a good pub if a general brawl didn't occur. A brawl during which chairs, bottles and other things fly fromone side of the pub to the other. Twoflower obviously documented everything with his camera. Now he is currently taking a picture of a ball of radius $Rthat$ is flying with a velocity $v$ (which is close to the speed of light $c)$. Even in such establishments the theory of relativity is valid and from it stems that Twoflower could have measured the Fitzgerald contraction of the ball in his rest frame in the direction of movement by a factor of

$$\\ \sqrt{1- \frac{v^2}{c^2}}$$

What radius of the ball was documented by the camera with a negligibly small exposition?The position of the camera is general.

Terka went on a trip once again. This time she was walking during equinox at twelve o'clock on the Equator. What would her velocity be relative to Ales, if he would want to (foolishly) watch her from the surface of the Sun on the Equator at a point nearest to his object of interest (Terka)? The axial tilt of the sun with respect to the plane of the ecliptic can be considered negligible.

Teresa was approaching with a relativistic speed $v$ a plane mirror. She was approaching perpendicularly to the mirror's plane. While doing so she is watching herself approach the mirror. What is the actual speed that she is approaching her image with and what is the speed she is observing?

Bonus: It isn't a plane mirror but a spherical one

By how much shall the temperature of two identical steel balls rise after their collision?They move in the same direction with speeds $v_{1}=0,7c$ and $v_{2}=0,9c$ where $c$ is the speed of light. Assume that the heat capacity is constant and that the balls are still solid.

What would be the world like if the speed of light was only $c=1000\;\mathrm{km}\cdot h^{-1}$ while all the other fundamental constants stayed unchanged? What would be the impact on life on Earth? Would it even be possible for people to exist in such a world?

• Any theory of quantum gravity is useful only when we deal with very small distances where the effects of gravitation are comparable to quantum effects. Gravitation is characterized by the gravitational constant, quantum mechanics by the Planck constant, and special relativity by the speed of light. Look up numerical values of these constants, and, using standard algebraic operations, combine them to obtain a quantity with the dimensions of length. This is the length scale where both quantum mechanics and gravitation are important.

• Prove that the special Lorentz transform (i.e. a change of the reference frame to one that is moving with speed $v$ in the $x¹;$ direction)

$$x^0_\;\mathrm{nov}=\frac{x^0-\frac{v}{c}x^1}{\sqrt{1-\(\frac{v}{c}\)^2}}\,,\quad x^1_\mathrm{nov}=\frac{-\frac{v}{c}x^0 x^1}{\sqrt{1-\(\frac{v}{c}\)^2}}\,,\quad x^2_\mathrm{nov}= x^2\,,\quad x^3_\mathrm{nov}= x^3$$ leaves the spacetime interval invariant. * Set $Δx=Δx=0$ in the definition of a spacetime interval. You should get

$$(\Delta s)^2 = -\(\Delta x^0\)^2 \(\Delta x^1\)^2$$

What is the region of the plane ( $Δx^{0},Δx¹;)$ where the spacetime interval ( $Δs)$ is positive? Where negative? What is the curve ( $Δs)=0?$