It would obviously utilize a much more powerful substance than simple electricity. Maybe plasma could constantly be siphoned from the infinitude of stars in the universe to keep the battery charged constantly?
This question was inspired by L. Ron Hubbard’s story “Revolt in the Stars”, which tells of an alien dictator named Xenu. He ended up being condemned for genocide on a cosmic level, and was locked away in a force-field prison. Said force-fields were powered by an eternal battery.
Yes. It's called a "betavoltaic" battery (there are also theoretical alpha- and gamma-voltaics devices, but betavoltaic batteries - or rectius, unreachargeable betavoltaic cells - are known to already exist).
To build an "eternal" battery, you'd use a very large quantity of 128Te (or any other "barely radioactive" element) surrounded by substances that react to its double beta decay by generating photons, and suitably doped semiconductor layers (essentially a photovoltaic cell). The tellurium will decay very slowly, supplying energy.
Since 128Te is unbelievably long-lived, the energy generated is minuscule. So, you will need a lot of tellurium.
On the other end of the spectrum, plutonium 238 decays in less than 90 years, and its activity (more than 600 GBq/g) and energy output is high enough to keep a nugget of PuO red-hot.
A large enough deposit of uranium ore should be enough to keep Xenu locked for the foreseeable life of most current generation's stars.
The quantity of material required is probably pretty much constant with the power requirement over the same period of time; that is, if you double the decay rate you do double the power, and require half the quantity, but then you need to double that again since otherwise it would burn in half the time.
From Wikipedia I get uranium 235's half life, about 700 million years. So if I have a kilogram of 235U, in 700 million years I'll have half a kilo, after 1.4 billion years 250g, after 2.1 billion years 125g, and so on.
Activity: 2.12 microCurie per gram. What does this mean: since one Curie is 3.7 x 1010 atoms per second, and one gram contains 2.12 microCurie, so 2.12 x 3.7 x 1010 x 10-6 = 78440 atoms per gram disintegrate each seconds. Each of those gives out 4.39 MeV, or 1.60217733 x 10-13 J.
78440 x 1.6E-13 J each second is 1.25 x 10-8 W, which is very little (this is spontaneous radioactivity; of course were we to enclose the metal into a neutron reflectant material, such as beryllium or tungsten carbide (you might want to google "Demon Core"), the radioactivity would increase sharply, and the half life go down accordingly.
Assuming an advanced enough technology we should be able to recover a good 50% of that energy, so we can extract 6E-9W from each gram, 6E-6W from each kilogram, 6 milliwatt from each ton, and 6 W from a block of one thousand tons. A million tons of U235 (divided into subcritical masses separated by neutron absorbers, to avoid uncomfortable nuclear explosions) would supply initially six kilowatts, down to 3 kW after 700 million years; and so on.
Given a density of around 19, one cubic meter of uranium weighs 19 tons, and one million tons of uranium are one cube with a side of 37 m (given the need of subdividing it into subcritical masses and gather energy, I imagine it would be more like 50 or 60 m).
If the field generator requires one hundred kilowatts (100 / 6 kW = about 17 million tons) and we want it to be running for twenty billion years (20,000 / 700 = 28 half-lives), we need (100/6)*228 = 4.5 billion millions tons, or about 220 thousand cubic kilometers; a sizeable asteroid of about 75 km diameter.
Such a mass would significantly distort the planet's local gravitational field, so if that's how the eternal battery works, I feel some confidence in telling you that good ol' Xenu might currently be doing time somewhere in the vicinity of Manaus, Brazil.
As user @Hobbes noted, the problem is not a battery running for billions of years - the problem is maintenance. The field generator, and the battery itself will degrade over time (all the more so since radioactivity plays hell with electronics and semiconductors). Even with self-recycling and self-repair, chances are that after a paltry few million years, Xenu is going to get free.
With the reality-check tag, there is no such thing. The first law of thermodynamics says that there are no perpetual motion (or perpetual power) machines. You tried to get around this by tapping stars, but even stars are not eternal.
I'd suggest a substellar mass black hole. Given that we're dealing with science fantasy silliness, the safe anchoring of the black hole to the planet is left as an exercise to the reader, but it'll probably help if your black hole is charged. Good luck!
Black holes evaporate over time due to Hawking radiation. This can be captured by various means such as photovoltaics or even a heat engine, given that you can use the rest of the earth as a heat sink.
Lets be less optimistic than LSerni, and assume the prison needs about a megawatt to run. The Hawking radiation power of a black hole with mass $M_0$ is $P \approx 3.56345\times10^{32} / M_0^2$, so if we need a megawatt we'll need a black hole with mass of 1.9x1013kg, or about 20 billion tonnes. This compares favourably with LSerni's millions of billions of tonnes, and provides more power and by way of a bonus black holes develop more power as they shrink... no half-life issues here.
A black hole with initial mass $M_0$ will evaporate over this timescale: $t_\mathrm{ev} \approx 8.41092 \times 10^{-17} \;M_0^3$. For our initial megawatt power source, we get a lifetime of 5.6x1023 seconds, or about 18 quadrillion years, vastly outperforming LSerni's and Gloweye's suggestions and when the black hole runs out of mass it goes bang spectacularly rather than just turning off and letting the occupant of your prison escape. Not that it will matter because the universe isn't guaranteed to last that long.
Aha! I suspect people will say. There's no way your prison equipment could last that long! They're probably right of course.
I'd just throw Xenu into the event horizon of a larger black hole (such as the regular, natural stellar-mass ones). If they're genuinely immortal (and they'd have to be, if imprisoning them for eternity is a worthwhile punishment) then I'm sure a little thing like spaghettification won't bother them too much and not much gets out of event horizons. Problem solved.
edit: to counter the "the black hole will eat the earth!" argument, consider than the event horizon of a black hole is found at the Schwarzchild radius which for this miniscule object will be significantly smaller than a single atom. It cannot eat the Earth quickly enough to make any difference to the lifetime of either the black hole or the Earth itself (which will of course be eaten by the Sun in the future). Have a read of this related answer for a bit more on the matter.
Obviously, the most sensible thing to do is no not drop the black hole, hence the usefulness of it being charged. You might need a lighter, more energetic black hole if you wanted to get enough energy to electromagnetically levitate it against Earth's gravity, but maybe you should just keep the whole assembly sensibly in space instead. The OP, of course, did not mention planets or gravity at all, merely eternity...
Since it's cosmic-scale stuff we're talking about, a star can last a pretty long time, and 10 billion years is very possible. The vast majority of that time (like 80% or something), it'll have a pretty constant output, much like our own sun.
And if you have FTL travel, then you can even abandon one star and head to the next if you're close to running out. It shouldn't be that much of a burden to do that once every X billion years.
It would obviously utilize a much more powerful substance than simple electricityThere seems to be a fundamental misunderstanding here. Batteries produce electricity. They store energy, usually in the form of a chemical reaction that proceeds when the terminals care connected. – nzaman Oct 21 '19 at 12:22