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Assume two ships accelerating uniformly beside each other, and in the same direction.Each one will have its own local (same) 'gravity', equivalent to the acceleration.Do those fields influence each other?

So what is it that 'slows' you down infinitely, as you come forever closer to lights speed in a vacuum?Relative mass is one expression, momentum another, what more?==

Sort of elegant Mike, although I need to think about it some more. A geodesic is indeed (a result of) 'gravity', and when objects following a geodesic is acted upon they find inertia. But then you have something accelerating, if acted upon that too will find a inertia, but it is not following a geodesic. The idea has a certain symmetry to me though, and SpaceTime seems very much to be about symmetries.

I think we could simplify this Question and instead of spaceships ask do highly accelerated particles in the LHC bunch due to gravitational attraction or do coulamb forces overwhelm this effect.

Now that's tricky Mike If we assume that inertia is a intrinsic property of matter you must be wrong. If we assume that inertia is 'the tendency of a body to maintain its state of rest or uniform motion unless acted upon by an external force' then you might formulate it that way?So, what is it?A intrinsic property, or a special circumstance?

Okay, I'm changing track here it seems clipSo, are we happy with defining a accelerating ships 'gravity' to its own 'intrinsic' properties, ignoring the mass, or do we think that two accelerating ships/particles will influence each other gravitationally to a higher degree the faster they move relative Eh, Earth It also have an importance for the view in which we imagine all accelerations as continuous displacements of a uniform motion, as it seems to me?

clipBut if I imagine something moving and I then stop it. Is it its inertia that comes to rule in a uniform motion? But not in a acceleration? I would call that pretty weird if it was so?

Assume two ships accelerating uniformly beside each other, and in the same direction.Each one will have its own local (same) 'gravity', equivalent to the acceleration.

Mike, we better ignore inertia for this. I think we can use momentum instead? Inertia is a description similar to rest mass in that it is a invariant local property of a object, expected to be the same everywhere, according to a main stream definition. You might say that the 'inertia' only is definable from a 'uniform motion' in relativity. Although I thought of it differently But using momentum does describe it as I think of it.==To see my point consider if different uniform motions will give you a different 'inertia'? When it comes to a local acceleration it won't as I see it. But if you're trying to stop two objects in a uniform motion, that you define as having different speeds relative you. Will the 'force' you apply be the same for making them 'unmoving' relative you?But I keep losing the track here So momentum will do for this I think?

So you know what inertia is then Pete