Btukfyl

Consider Earth moving around the Sun. Is the force of gravity exerted by Earth onto the Sun directed towards the point where Earth is "right now", or towards the point where Earth was 8 minutes ago (to account for the speed of light)?
If it's the former, how does the Sun "know" the current orbital position of Earth? Wouldn't this information have to travel at the speed of light first?

If it's the latter, it would force a significant slowdown of Earth's orbital motion, because the force of gravity would no longer be directed perpendicular to Earth's motion, but would lag behind. Obviously, this isn't happening.

So it appears that the force of gravity is indeed directed towards the current orbital position of Earth, without accounting for the delay caused by the speed of light. How is this possible? Isn't this a violation of the principle that no information can travel above the speed of light?

Cuckoo asked: So it appears that the force of gravity is indeed
directed towards the current orbital position of Earth, without
accounting for the delay caused by the speed of light. How is this
possible?

If the motion is straight or circular the aberration cancels out, see Steve Carlip: Aberration and the Speed of Gravity:

Steven Carlip wrote: The observed absence of gravitational
aberration requires that "Newtonian'' gravity propagates at a speed
ς>2×10¹⁰c. By evaluating the gravitational effect of an accelerating
mass, I show that aberration in general relativity is almost exactly
canceled by velocity-dependent interactions, permitting ς=c. This
cancellation is dictated by conservation laws and the quadrupole
nature of gravitational radiation.

or to quote the Wikipedia article on the subject:

Wikipedia wrote: Two gravitoelectrically interacting particle
ensembles, e.g., two planets or stars moving at constant velocity with
respect to each other, each feel a force toward the instantaneous
position of the other body without a speed-of-light delay because
Lorentz invariance demands that what a moving body in a static field
sees and what a moving body that emits that field sees be symmetrical.
In other words, since the gravitoelectric field is, by definition,
static and continuous, it does not propagate.

No, gravitational influences never travel faster than the speed of light. However, a naive incorporation of a speed-of-gravity delay would actually lead to the Earth's orbital motion speeding up, not slowing down. (Think about the geometry carefully.) I explained here why that doesn't actually happen in general relativity.

Think of the gravitational field as curved space. Like a bump in space time. This bump will stay constant over time more or less. The planet then just looks at the bump and moves accordingly. However if the sun were to explode it might cause a small ripple in spacetime that would reach the earth in 8 minutes

I was going to write an incorrect answer at first, but having researched a bit
what Yukterez wrote, I think I may be able to offer some intuition.

First, let's look at the solar system in the centre-of-mass frame, where the
Sun is essentially stationary in the middle. Here, the force on the Earth is
directed towards the Sun's position 8 minutes ago, which conveniently is the
same as its current position. So this doesn't affect the Earth's position in
the way you say.

If the Sun and Earth were moving at a constant velocity to each other, they
would, like Yukterez wrote, still be attracted to the instantaneous position of
the other body. This is not because of a faster-than-light influence, but
because gravity is more than a simple attractive force between objects
(analogous to the electric field in electromagnetism). For moving sources,
there are other components of the gravitational field (roughly analogous to the
magnetic field). You can view this as the force travelling at the speed of
light, but being directed towards the position of the source "predicted" based
on its past linear motion.

However, if the source is accelerating, like the Earth around the Sun, the
effects are less obvious. Then, you really can't view it only as an attractive
force towards any position (I think, correct me if I'm wrong), but you get
more complex things like gravitational waves. But still, nothing ever travels faster than
light.

As I am more familiar with electromagnetism, let's take an example from there.
If two oppositely charged bodies ($A$ and $B$) are moving with constant
velocity towards each other, we can view the system in the rest frame of $A$.
Here, it emits only a static electric field, given by the Coulomb formula, and
$B$ therefore feels an attraction directly towards it. In the rest frame of $B$,
this means that the attraction is towards the instantaneous position. But how
can this be since the electric field is propagating at a finite speed? The
answer is that since now $A$ is moving, it also emits a magnetic field. While
this doesn't affect $B$ directly (since $B$ is stationary here), it does affect
the electric field and makes it point in a different direction (towards the
instantaneous position).