Do the forces between Earth and Sun point in the directions that the Earth and the Sun were 8 minutes ago?

Question

No information can travel faster than light, so does the force between Earth and Sun point in the direction that the Earth was 8 minutes ago or in the direction that the Earth is now?

Answer 1

… so the force beetwen Earth and Sun points in the direction that the earth was 8 or points in the direction that the earth is now?

No. Gravitational acceleration is directed towards the position of the source quadratically extrapolated from its “retarded position” toward its instantaneous position (that is using velocity and acceleration from that moment in a truncated Taylor expansion for the current position), up to small nonlinear terms and corrections of higher order in velocities.

No information can travel faster than light …

This does not mean that the force must be directed toward retarded position of the second body since the “information” transmitted may also include velocity and acceleration.

In nonrelativistic situations, such as motion of Earth around the Sun, this “quadratic extrapolation” matches the instantaneous position with a high degree of precision. For Earth–Sun system the discrepancy (which could be approximated by the next term of Taylor expansion $\approx \left|\frac{1}{6}\frac{d^3 \mathbf{r}}{dt^3}\,\tau^3\right|$) would be about several centimeters. Whereas discrepancy between instantaneous and retarded positions is about $15\,000\,\text{km}$. So the effects of such “gravitational aberration” are nonexistent within the Solar system.

One should keep in mind that all discussions of finite speed for gravity propagation must proceed within the framework of general relativity which has inherent ambiguity in identifying concepts of Newtonian physics such as force direction. While in Newtonian physics gravitational force between isolated bodies is defined unambiguously in GR we instead have the connection ($\nabla$) and curved spacetime. In order to extract from it Newtonian force and position of the bodies we must choose a specific reference frame, and impose a gauge conditions, which provides a certain freedom in the definitions.

Also note, that for the somewhat similar situation of two charges orbiting each other the direction of electrostatic force is given not by quadratic but by linear extrapolation from retarded position (using only velocity and not acceleration). The difference is in dipole character of EM radiation versus quadrupole nature of gravitational radiation.

For more information see the following (rather technical) paper:

Carlip, S. (2000). Aberration and the speed of gravity. Physics Letters A, 267(2-3), 81-87, DOI:10.1016/S0375-9601(00)00101-8, arXiv:gr-qc/9909087

Answer 2

This is more complicated than it first looks. Two forces. The force from the sun on the earth, and the force from the earth on the sun.

The force on the sun from the earth -- which has rather small effect -- has three components. There is the force that the earth would have had if it was stationary in our frame. Second there is the force due to the velocity of the earth 8 minutes ago. These two can be added to give a force that comes from the direction the earth would be now, if it had continued to travel at constant velocity for 8 minutes.

The third force, due to the earth's acceleration, appears to come not from where the earth would have been now, but from where the earth actually was 8 minutes ago.

None of these have a direct relation to where the earth is now, since in those 8 minutes many other forces could act on the earth.