Question

The star Alpha goes supernova. Ten years later and 100 ly away, as measured by astronomers in the galaxy, star Beta explodes.

a. Is it possible that the explosion of Alpha is in any way responsible for the explosion of Beta? Explain.

b. An alien spacecraft passing through the galaxy finds that the distance between the two explosions is 120 ly. According to the aliens, what is the time between the explosions?

Short answer

The explosion of alpha is not responsible for the explosion of beta. According to the aliens, the time between the explosions is 67.1 years.

Step-by-step solution

Given information

We have given that, the star Alpha goes supernova. Ten years later and 100 ly away, as measured by astronomers in the galaxy, star Beta explodes. An alien spacecraft passing through the galaxy finds that the distance between the two explosions is 120 ly.

Common information for both parts

Let S be the galaxy's frame of reference and S' be alien spacecraft's frame of reference. The spacetime interval between the two events is invariant in all frames.

Part(a) Step 1.

The light from Alpha's explosion will travel 10ly in 10 years.

Step 2.

Since neither light nor any other signal from alpha can travel 100ly in 10 years to reach beta, the explosion of alpha could not cause the explosion of beta.

Part(b) Step 1.

The spacetime interval between two events is invariant, so :

$s=c^2{\left(\Delta t\right)}^2-{\left(\Delta x\right)}^2=c^2{(\Delta t\text{'})}^2-{(\Delta x\text{'})}^2$

Step 2.

Substituting the values in equation :

${\left(\frac{11y}{y}\right)}^2{(10y)}^2-{(100ly)}^2={\left(\frac{11y}{y}\right)}^2{(\Delta t\text{'})}^2-{(120ly)}^2$

${(10y)}^2-{(100y)}^2={(\Delta t\text{'})}^2-{(120y)}^2$

$\Delta t\text{'}=67.1years.$