Swimming in spacetime: using your own gravity field? It is known that in a curved spacetime, due to the center of mass not beeing clearly defined object can change position using contortions. However an object generates its own gravity and changes in spacetime propagate only with lightspeed. So could an object swim in its own field if it was   fast enough?
 A: Is suspect this is not the scenario you were thinking of, but there is a trivial sense in which you can:
If the object changes its mass quadrupole moment (or any of its higher mass or spin moments), it will emit gravitational waves. Besides energy, gravitational waves also carry linear momentum. If the object changes its multipole moments in a sufficiently asymmetric way, more linear momentum will be emitted in certain directions than in others. Consequently, there will be a net emission of linear momentum in one direction, resulting in a net thrust on the object.
A well-known case of this happening is during the merger of a black hole binary, where in certain configurations the net "kick" to the system can be thousands of kilometers per second. However, in principle, the same effect could be achieved with an artificial generator of gravitational waves.
In terms of local dynamics, this effect takes to form of an effective force on the object due to interactions with its own gravitational field, a so called gravitational self-force.
However, I suspect you were looking for a scenario where there is no flux of momentum to infinity, but only purely conservative dynamics. In this case, the well-known mechanisms for "swimming in spacetime"  do not work in flat spacetimes (since they rely on couplings between the multipole moments of the object to the background curvature.)
I am not sure if there can be purely conservative effects due to the multipole moments interacting with the object's self-field in flat background. The mass monopole's contribution to the effective self-field vanishes in a flat background, so if the effects exist they would have to be higher order.
A: Your center of mass would not change its position or its velocity, relative to an outside observer, no matter how you moved. Moving your arms or legs one way would move the rest of your body the other way per action and reaction, but when you pull them back to their original position you will have the same action and reaction in reverse with the COM unmoved.  You may move your COM to a different point of your body but its position or velocity, relative to an outside observer will not change, he will see your body moving back and forth around a  COM that does not wobble.
