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If we (the galaxy) were traveling close to the speed of light; relativity says we would need proportionally more energy to go faster. Given that relative to the cosmic microwave background, the Local Group is moving by 600 km/s; that's 0.2% of the speed of light. Is it possible to look at this in terms of mass/light speed.

i.e.light speed =infinite mass.
.2% light speed =x mass. stationary = mass. x = speed

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No, in our own frame (comoving frame) we are not moving at all. Even though the CMB radiation is defining a preferred frame, to us it looks like the CMB is moving with respect to us - that's why we see a kinematic dipole in the CMB maps: http://scienceblogs.com/startswithabang/2013/06/28/our-great-cosmic-motion/

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