Subfield: Modern Physics

- Proton mass, $m_{p}=1.67 \cdot 10^{-27} \mathrm{~kg}$
- Neutron mass, $m_{n}=1.67 \cdot 10^{-27} \mathrm{~kg}$
- Electron mass, $m_{e}=9.11 \cdot 10^{-31} \mathrm{~kg}$
- Avogadro's constant, $N_{0}=6.02 \cdot 10^{23} \mathrm{~mol}^{-1}$
- Universal gas constant, $R=8.31 \mathrm{~J} /(\mathrm{mol} \cdot \mathrm{K})$
- Boltzmann's constant, $k_{B}=1.38 \cdot 10^{-23} \mathrm{~J} / \mathrm{K}$
- Electron charge magnitude, $e=1.60 \cdot 10^{-19} \mathrm{C}$
- 1 electron volt, $1 \mathrm{eV}=1.60 \cdot 10^{-19} \mathrm{~J}$
- Speed of light, $c=3.00 \cdot 10^{8} \mathrm{~m} / \mathrm{s}$
- Universal Gravitational constant,

$$
G=6.67 \cdot 10^{-11}\left(\mathrm{~N} \cdot \mathrm{m}^{2}\right) / \mathrm{kg}^{2}
$$

- Solar Mass

$$
M_{\odot}=1.988 \cdot 10^{30} \mathrm{~kg}
$$

- Acceleration due to gravity, $g=9.8 \mathrm{~m} / \mathrm{s}^{2}$
- 1 unified atomic mass unit,

$$
1 \mathrm{u}=1.66 \cdot 10^{-27} \mathrm{~kg}=931 \mathrm{MeV} / \mathrm{c}^{2}
$$

- Planck's constant,

$$
h=6.63 \cdot 10^{-34} \mathrm{~J} \cdot \mathrm{s}=4.41 \cdot 10^{-15} \mathrm{eV} \cdot \mathrm{s}
$$

- Permittivity of free space,

$$
\epsilon_{0}=8.85 \cdot 10^{-12} \mathrm{C}^{2} /\left(\mathrm{N} \cdot \mathrm{m}^{2}\right)
$$

- Coulomb's law constant,

$$
k=\frac{1}{4 \pi \epsilon_{0}}=8.99 \cdot 10^{9}\left(\mathrm{~N} \cdot \mathrm{m}^{2}\right) / \mathrm{C}^{2}
$$

- Permeability of free space,

$$
\mu_{0}=4 \pi \cdot 10^{-7} \mathrm{~T} \cdot \mathrm{m} / \mathrm{A}
$$

- Magnetic constant,

$$
\frac{\mu_{0}}{4 \pi}=1 \cdot 10^{-7}(\mathrm{~T} \cdot \mathrm{m}) / \mathrm{A}
$$

- 1 atmospheric pressure,

$$
1 \mathrm{~atm}=1.01 \cdot 10^{5} \mathrm{~N} / \mathrm{m}^{2}=1.01 \cdot 10^{5} \mathrm{~Pa}
$$

- Wien's displacement constant, $b=2.9$. $10^{-3} \mathrm{~m} \cdot \mathrm{K}$
- Stefan-Boltzmann constant,

$$
\sigma=5.67 \cdot 10^{-8} \mathrm{~W} / \mathrm{m}^{2} / \mathrm{K}^{4}
$$

Question: Pegasi and Betelgeuse are two star systems that can undergo a supernova. Betelgeuse is 548 light-years away from Earth and IK Pegasi is 154 light-years away from Earth. Assume that the two star systems are 500 light-years away from each other.
Astronomers on Earth observe that the two star systems undergo a supernova explosion 300 years apart. A spaceship, the OPhO Galaxia Explorer which left Earth in an unknown direction before the first supernova observes both explosions occur simultaneously. Assume that this spaceship travels in a straight line at a constant speed $v$. How far are the two star systems according to the OPhO Galaxia Explorer at the moment of the simultaneous supernovae? Answer in light-years.

Note: Like standard relativity problems, we are assuming intelligent observers that know the finite speed of light and correct for it.

Solution:
For any inertial observer, define the 4-distance between two events as $\Delta s^{\mu}=(c \Delta t, \Delta \mathbf{x})$, where $\Delta t$ and $\Delta \mathrm{x}$ are the temporal and spatial intervals measured by the observer. By the properties of 4 -vectors, the following quantity is Lorentz-invariant:
$$
\Delta s^{\mu} \Delta s_{\mu}=c^{2} \Delta t^{2}-\|\Delta \mathbf{x}\|^{2}
$$

In the spaceship's frame, this is equal to $\Delta s^{\mu} \Delta s_{\mu}=-\left\|\Delta \mathbf{x}^{\prime}\right\|^{2}$, since the two supernovas are simultaneous; hence

$$
\left\|\Delta \mathbf{x}^{\prime}\right\|=\sqrt{\|\Delta \mathbf{x}\|^{2}-c^{2} \Delta t^{2}}
$$

Since the observers have already taken into account the delay due to the nonzero speed of light, $c \Delta t=300 \mathrm{ly},\|\Delta \mathbf{x}\|=500 \mathrm{ly}$, and

$$
\left\|\Delta \mathbf{x}^{\prime}\right\|=\sqrt{\|\Delta \mathbf{x}\|^{2}-c^{2} \Delta t^{2}}=400 l y
$$

Final answer: 400 light-years