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General relativity assumes that space itself is warped by gravity, which is what gives rise to the shapes of the paths followed by objects subject to a gravitational field.

If space itself truly bends, then the paths of all particles in an area of space warped by a gravitational field should be affected.

As a result, the path of the force carrier of an electrostatic field should also be affected by the presence of a gravitational field, since the path of that force carrier should also be affected by the local warping of space.

Therefore, any electrostatic field that is within an area of space warped by a gravitational field should in some way be affected the presence of the gravitational field.

If so, has the effect been measured by experimentation?

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You are correct, as per GR we usually use the phrase spacetime is bent by gravity.

Now you are saying that the path of the force carrier of an electrostatic field should be bent too.

Now in the case of the electrostatic field, we use virtual photons to describe the interaction between the field an another particle it interacts with.

These virtual particles are not real, they are just a mathematical model to describe the field.

These virtual particles should not be affected by bent spacetime.

Though, the electric field lines of the electrostatic field in your case, will follow the same trajectory as light, which is made up of real photons.

Therefore, the electrostatic field's shape is affected by bent spacetime. This means that the field lines will follow the same trajectories as real particles.

https://link.springer.com/chapter/10.1007%2F978-3-319-13443-7_26