[geordief (OP)]

 I have heard that we are all moving wrt to the centre of the Earth and the Earth is in motion around the Sun  etc etc.
 
Are there places in the Solar System that share the same rest frame(other than by reason of being adjacent on a fixed body such as a planet or moon?

If such places exist in the Solar System ,what about in the Galaxy?

Not absolutely at rest (I want to be reasonable ) but as close as possible.

[jeffreyH]

The earth itself isn't in a rest frame so the question is moot. It would be considered to be at rest in an inertial frame if no forces were acting upon it. Since it rotates and orbits both the sun and the galaxy you would be hard pressed to find an object in an identical state of motion.

[geordief (OP)]

Thanks. I think I will take a rest now

[Bill S]

It would be considered to be at rest in an inertial frame if no forces were acting upon it.

As the Earth is orbiting the sun, it is not travelling in a straight line, so would it not be accelerating?
If this is the case, would that mean that it was not in an inertial frame?

[jeffreyH]

It would be considered to be at rest in an inertial frame if no forces were acting upon it.

As the Earth is orbiting the sun, it is not travelling in a straight line, so would it not be accelerating?
If this is the case, would that mean that it was not in an inertial frame?

That was the point I was attempting to make.

[Petrochemicals]

It would be considered to be at rest in an inertial frame if no forces were acting upon it.

As the Earth is orbiting the sun, it is not travelling in a straight line, so would it not be accelerating?
If this is the case, would that mean that it was not in an inertial frame?

It is travelling in a straight line isnt it? With the curvature of space time

[geordief (OP)]

It is travelling in a straight line isnt it? With the curvature of space time

If treated according to the Newtonian model it undergoes acceleration.

And the Newtonian model is extremely accurate for the purposes of the OP. (even though perhaps I did couch the question  in spacetime terms by the use of the phrase "rest frame" )

[jeffreyH]

If an object is following the curvature of spacetime then it cannot be travelling in a straight line. A straight line can be represented by a geodesic, a great circle, on a sphere. However this precludes drawing parallel lines due to the curved surface of the sphere. That is something entirely different to a straight line path in flat spacetime.

[geordief (OP)]

Is the definition of a straight line a question in itself?

Can there be different definitions depending on models or situations.

Is there one definition  that can cover them all?

[jeffreyH]

This is where the principal of least action comes in.
https://en.m.wikipedia.org/wiki/Principle_of_least_action
This ties in with your question on the definition of a straight line (shortest path between two points). In curved space-time this is a geodesic. A geodesic can be thought of as a straight line placed onto a curved surface. A large enough sphere, such as the earth, will be locally flat. So a straight line will not appear curved. Globally, say from the ISS, the curvature is very apparent. The gravitational field varies with distance from the surface and its gradient takes on a spherical character. With density variations altering this.

It is thought that lunar density variations, causing gravitational anomalies, may have doomed some probes sent to orbit the moon. The curvature of spacetime in action.

[guest4091]

In the real world, we can assume the earth and moon at relative rest for the 3 second experiment of reflecting a radar signal from the reflectors on the moon surface.

[kazbert]

Perhaps the OP is thinking of Lagrangian points?

[geordief (OP)]

Perhaps the OP is thinking of Lagrangian points?

I wasn't but they did occur to me after a while. Indeed a satellite positioned at one of these points would seem to be stationary with respect to their parent bodies..

[yor_on]

"It is travelling in a straight line isnt it? With the curvature of space time"

Yes, if you by it mean that it is having a uniform motion in a geodesic. Such a (ideal) one is defined as offering no resistance and no friction, which then allows it to go on 'for ever'. But gravity acts and are acted upon by other gravity, and loses 'energy' thereby.

It's a interesting question.

[geordief (OP)]

"It is travelling in a straight line isnt it? With the curvature of space time"

Yes, if you by it mean that it is having a uniform motion in a geodesic. Such a (ideal) one is defined as offering no resistance and no friction, which then allows it to go on 'for ever'. But gravity acts and are acted upon by other gravity, and loses 'energy' thereby.

It's a interesting question.

Does the "energy loss" depend on the frame of reference chosen ?