Unexpected Finger's engines produced 3.6 billion megawatts. How much of that power blew out her tailpipe toward Earth?
The glib answer is, all of it. This is wrong, but not for the reason you think. Finger blew a constant 3.6 billion megawatts out her tailpipe, and used some of that power to decelerate. The fun part is, the more of her exhaust energy she used to decelerate, the more she blew at us.
Rocket science is fun, but Conservation of Energy takes a beating.
Before we dive into the math, let's acknowledge Finger was moving pretty fast. Must we involve Dr. Einstein and worry about our clocks lagging, our rulers shrinking, our math cluttering up with greek letters, and our spouses contriving to gain weight while they flatten?
Let's check. Relativistic motion distorts time and space by the Lorentz factor, γ = 1/(1-v2/c2)½. Plugging in v=0.2c gives γ = 0.02.
In other words, Finger's velocity was only semi-relativistic, distorting time and space by two percent, which I will ignore because I don't want to be accused of pedantry. People already suspect, but I swear those are just fatherly hugs.
Back to Finger's exhaust, which was interesting because she was moving toward us, boosting her exhaust energy by Doppler shift.
Finger's exhaust was photons. When photons are Dopplered up to higher energy, we say they are blue shifted, and the shift increases with velocity. The relation between velocity and photon energy goes as c/(c-v), not the usual v2, because the blue-shifted photons do not actually move faster. Think of the laser chasing the departing photon, jamming in more energy before the photon gets away. In this case, the energy boost is c/(c‑0.2c)=1.25, a 25% increase.
Is this the part where we get to violate conservation of energy? How can we on Earth see more photon energy than Finger actually emitted?
To answer that question we must ask where, exactly did the photon get its energy?
Well, the photon was made from a proton, with no neutrino waste, so the photon carried all the parent proton's mass-energy. Plus the photon got some of Finger's kinetic energy, because when the photon left, Finger slowed. (That was the whole point.) Finger's kinetic energy had to go somewhere, and the photon was the only available receptacle.
Yeah I know, not a satisfying explanation. But get this: All the useful energy in Finger's exhaust (that is, all the energy used for braking) came from the energy each photon received from Doppler shift.
If you doubt me, try looking at it from your perspective.
From Earth, you saw Finger converting the mass of her fuel into the energy of her exhaust photons. The photons captured all the mass-energy of her fuel, with none left over to decelerate the ship. The trick was, the Finger was also converting her own kinetic energy into photon energy, slowing herself down in order to Doppler up her exhaust.
If you don't think that is cool, you would not be reading this paragraph.
We are still talking about blue shift, which most textbooks imply is the whole Doppler story. It is not. For you true geeks who were angered when Frank claimed a fifty-percent boost from Mr. Doppler at a velocity of only 0.2c, prepare to be mollified.
You will be mollified, albeit reluctantly, because Doppler shift also reduces the spacing between photons, further boosting Finger's apparent exhaust power.
Finger's exhausted photons raced away from her, while she pursued at twenty percent of their speed. So five seconds later, the photons would have covered five light-seconds but Finger would be only four light-seconds behind, spewing out more photons. Equivalently, five seconds worth of her exhaust was jammed into a four-light-second space. When that exhaust reached Earth, it would hit you with five seconds worth of photons in only four seconds, boosting the Finger's apparent exhaust power another 25%.
If you have studied physics, you may be feeling stress, because the second component of Doppler shift (photon spacing) was left out of your textbooks. It may help to imagine Finger was manned by panicked hippies, convinced they were about to destroy the Earth, trying to slow their progress by ripping off engine parts and flinging them at you.
Can I stitch a metaphor, or what?
Since Finger was moving toward you, its velocity would be added to each flung engine part, so each part would hit you harder. (That is blue shift.) And the parts all hit you in a shorter span of time. They had to, because they could not start hitting you until they had crossed Finger's entire braking distance. To got all those hits in before Finger's arrival, they had to hit you more rapidly than the hippies flung them out.
But wait, if Finger appeared to have more exhaust power, wouldn't she also appear to decelerate faster?
Yes, and she would need to, because she would appear to be moving faster, and appear to have less time to decelerate. But that is a story for a longer book. If you want to know more, start by googling Superluminal Jets.
But how could you and the crew disagree about something so basic as photon spacing? Were you afflicted by that relativistic contraction of space-time we hear so much about?
No, but but you did experience relativity.
From the crew's perspective, they were stationary and you were racing toward them. So they fired their engine to match your velocity before you and your planet rammed them. The crew even agreed that their exhaust photons hit you harder, and more often. But to them, it was because you were racing forward into the collisions.
This is what people mean, or at least what they should mean, when they say everything is relative.
Probably you are feeling stress, because this relativity stuff seems all about optical illusions, talking about how photons appeared to us, as if it mattered.
Yeah, that bugs me too.
To sum this up, Doppler shift has two components: photon energy (I.e., blue shift) and photon spacing. In numbers, Finger's exhaust power (as measured by the crew) was 3.6 billion megawatts. Her forward motion blue-shifted each photon up to 25% more energy, and Dopplered her total exhaust power up another 25% by crowding the photons closer together. So her total exhaust power (to an Earth observer) was 3.6×1015 W × 1.25 × 1.25 = 5.6×1015 W.
Six billion megawatts, near enough.
We saw half again the exhaust power measured by Finger's crew. Did that extra power come from relativity? Mystic dilation of space and time?
No, just Doppler shift, worked out by Mr. Doppler 37 years before Einstein's birth.