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On earth gravity stays constant (besides negligible fluctuations), but it may be possible to change that. What kind of geology, natural satellite, or other anomaly could cause periodic changes in gravity? (And what magnitude of difference would they have). The ideal goal is to map the change to a kind of sine wave, so that the bounds of the changes stay constant. The periodicity of the planet's gravity can range from days to years.
The gravity on Earth is changing. We see it in the tides of our oceans. The tides are caused by the Moon and the Sun which exert a gravitational pull on all the mass of our planet, but exert a greater pull on the parts of our planet closest to them because gravitational attraction decays with distance. The Moon isn’t very massive and the Sun is a long way away so on Earth the tidal forces aren’t very large, but if our planet were to orbit something very large and very close these tidal effects could become significant. If a moon orbits a large enough planet too closely, the gravitational gradient can overcome the gravitational self-attraction of the satellite and cause it to disintegrate. This distance is called the Roche limit. A body near the Roche limit will have reduced gravity on the sides facing directly towards and directly away from the body it orbits, but will have normal gravity elsewhere.
Any bodies in orbit will exert tidal forces on one another. These forces can range from nearly imperceptible, as on Earth, to making anything on one side of the planet that isn’t held down float off into space. The magnitude of these forces will depend on the size and density of both bodies and the distance between them. As long as the planets are not tidally-locked (rotating so only one side of the planet faces the other as our moon does) the gravity will change periodically, as the tides change on Earth. Additionally, an eccentric orbit could also introduce changes to these effects with the desired periodicity.
A very massive planet with an extremely high rotation speed would have an huge gravity gradient between the poles and the equator. The classic sci-fi book "Mission of Gravity" by Hal Clement exports such a world.
If your planet had a stable micro black hole orbiting inside it (well below the mantel) gravity would vary by location depending on the orbit of the black hole. Of course, such a planet would eventually be doomed, as the black hole would absorb more and more mass until it had absorbed enough to do catastrophic damage to the planet.
Just how a black hole would form a relatively stable orbit within a planet is left as an exercise for the reader, but unexpected results from a super-collider might be a good start.
My best guess would be some how an ultra dense piece in the mantle is circulating in the planet. Say a (very large) clump of lead (or even something more dense) that from some reason is moving around in the mantle of the planet. As it orbits the core, it changes the 'center' of gravity and affects gravity on the surface.
I would expect if this is even possible, that it would in reality be a very slow rotation, maybe once in a humans lifetime or once in a thousand years for a circuit to complete, but maybe a very active planet might have it more often, and that might be why it's a lot more active, say something large collided with the planet and everything hasn't completely settled down several million years later.
For the exact formulation you'd need a lot of maths and simulation, but if you only look for a feasible explanation, what Mike says is the easiest way to explain a changing gravity.
Big orbiting planets can stay at equilibria even when their orbits go one across the other, even when planets are almost nex to each other for a period of time. Therefore, you could explain the changes on gravity that way: When another orbiting planet goes nearby, gravity is way stronger towards that planet. Also, a big "star" (it's a planet, not a star, but people might be unable to differentiate) appears on the sky for that duration of changed gravity, which is really interesting from a mythological aspect.
If you want even more variation, add more planets to the "equation". That way, the mythology around that goes even deeper. Every planet, with it's distinguishable bright and color (since star's light are reflected differently) brings a different gravity and for a different duration of time.
How about the following.
We could postulate the core has absorbed a particularly large incident body some time back, that for some reason has never quite migrated to its geometrical center.
I am not saying this situation would be stable forever, but why not over a geologically insignificant and biologically large period, like 10 million years?
Along similar lines to the black hole scenario, you might have some weakly interacting dark matter orbiting inside the planet.
You need the dark matter to clump together and to have minimal interaction with normal matter.
A candidate for this would be Weakly Interacting Massive Particles. Wikipedia references a paper suggesting they would tend to clump together - I am not sure how well validated this is, but good enough for a story.
Note we still have to worry about the Roche limit, whether inside or outside the planet. I think that if it is dense enough, it will be stable.
A easy and fun way to do it is to place several moons orbiting your planet. They have different trajectories in space, for example almost comet like for one, thus coming only once a decade. The other moons could be orbiting in whatever way you want.
I would keep it simple but if you are ready to push yourself, nothing better than the usual duo planet orbiting one star with a moon or 2 to gear things up. There you got retard gravity.
A few of the answers here propose a very high-density mass located off-center in the planet's core, along with the different rotational period of the core to the surface, leaving the problem of how this mass came to be there in the first place. What about the opposite scenario?
A moderately-sized cavity in the planet's solid core, off-center - which could either be filled with a significantly lower-density material, (helium perhaps?), or simply be a vacuous void in the core. As with the other explanations, with the core's rotational period differing from that of the surface thanks to the liquid outer core between them, the position of the planet's center of mass would shift relative to the surface geography. But a region of lower density material is, perhaps, easier to explain than one of highly dense material - a mining operation, a chemical reaction (the other reactant being less abundant then the core material, it eventually ran out), or even being scooped out by a rogue wormhole.
In a universe with something like Tensor–vector–scalar gravity, a Type II civilisation (in its early days) created wormholes or warp drive. Sean Carroll has a few youtube videos that cover Dark Matter: one in particular is more in-depth for TeVeS, but I can find it right now. He uses the less-technical version in several talks.
I find it interesting that TeVeS implies a preferred reference frame for gravity— exactly my favorite way to make FTL travel safe for the arrow of time. He explains how the Ve and Te fields would allow interesting gravitational phenomena to exist (if we can find a way to express it).
So, it all fits: a wormhole hub ruins or an "accident" leaves behind something that makes the additional fields mess with gravity. That civilisation is long gone, and new intelligences are dealing with it. Or maybe it's current and some people have to put up with that as, say, an inherent part of doing business at a major shipping hub.
Expanding on the satellite solution and the issue with the Roche limit.
The Roche limit goes as the cube root of the relative densities. If $M$ refers to the planet and $m$ to the satellite:
$d_{Roche} = R_M(2\rho_M/\rho_m)^{1/3}$
It follows that if the satellite is twice the density of the primary, the Roche limit is not a problem.
So, make your primary have low density and the satellite nickel-iron.
For added amusement:
It is really a planet?
I'm thinking of David Weber's Mutineer's Moon. (There are other stories in the series but this is the relevant one.) It turns out Earth's moon was actually destroyed 50,000 years ago--what we see in the sky is a starship on long term picket duty that was concealed by peeling off the outer layers of our moon and covering the ship with them. There was a mutiny, the ship was severely damaged and remains there to this day inhabited only by the now-sentient computer core.
The gravitational anomalies we see on the moon are really the result of the mass distribution of the underlying starship. While it does nothing that changes the gravity of the moon it certainly would be capable of doing so.
If you want to play around with what's going on inside your planet, feel free. Gravity is defined as $F=\frac{Gm_1m_2}{r^2}$, where $m_1$ and $m_2$ are the masses of the objects in question, $G$ is a constant, and $r$ is the distance between the two objects. Since $G$, $r$, and $m_1$ are all constant, you can have the inside of the planet change its mass by some chemical reaction (unobtanium?). As $m_2$ increases, $F$ increases; as $m_2$ decreases, $F$ decreases.
Well, with gravity you mean probably gravitational pull at the surface. So there are some possibilities:
And naturally: the world simply has different laws of physics.
It is hard to think about a scenario how the planet loses and gains mass. Something I think of: the planet has an enormous part of its mass in the form of water, that is stored underneath the crust. Some events lead to evaporation of it, so the main mass of the water is leaving to higher atmosphere and raining later back to the planet. Still complicated.
The other possibilities are also difficult. Expansion and contraction of the crust in big style is something I have no idea about how it could happen without breaking suspension of disbelief. Same thing about changes in rotation speed. Some big mass (say a pretty heavy moon) might sound believable, but it seems something that big induces strong tidal forces that will lead to reduction of rotation speed around each other fast (as happens with the moon of the earth, it shows already the same side to the earth at all times, because of the tides induced from earth on the moon and the tides the moon induces on earth will other time lead to the moon being at the same place over the surface of earth at all times).
So I would go with a world of different physics. The force of gravitation is changing regularly.
