Before I tackle the real puzzle, let's clear up some minor errors. If "anything" refers to a grain of sand, for instance, it probably won't go straight thru your space ship; instead, it will explode at the surface like a small atomic bomb and vaporize a large part of the ship.

I disagree with the statement that, "...the background radiation of the universe will be blue-shifted up to very uncomfortable temperatures." Temperature is inversely proportional to wavelength. To heat the CMBR in front to 0°C, you would need a relativistic Doppler shift of 273 K / 2.7 K = 101.1, which corresponds to a relative speed of 299734 km/s = .9998 c. (To derive that, I went to Wolframalpha, input "relativistic Doppler shift", select "calculate speed of source away from observer" and "frequency reduction factor = 101.1". Since the reduction factor is greater than 1, the result is negative 299734 km/s. Click on that answer to convert to .9998 c.)

Now for the real brain buster! I'll have to solve the more general question of what the CMBR would look like to an observer moving at .97 c relative to practically all of the matter in the observable universe. I've been working on this, off and on, for a week.

Trying to solve this question has raised a number of other questions that puzzle me. Perhaps someone here can help. If you don't understand

comoving coordinates, you won't understand my argument, so do some boning up, first. The increasing distance between distant galaxies is not considered to be a real velocity in comoving coordinates, so it does not contribute to the length contraction or time dilation of special relativity.

The key to solving this problem is in setting up two comoving coordinate systems, with relative motion between them at .9682 c in the +x direction, which corresponds to gamma = 4. They are comoving within themselves, but not comoving with each other.

One system, F0, is comoving relative to Earth. Since known galaxies have only minor proper motion (compared to .9682 c), we may consider that all galaxies in the visible universe are comoving with Earth. In other words, the only significant motion between us and the galaxies, in F0, is the expansion of space, and all galaxies are approximately motionless relative to their coordinates in this expanding coordinate system.

The other coordinate system is comoving relative to imaginary points in space, which are all moving at .9682 c relative to the corresponding points in F0. All galaxies are moving thru F1 at .9682 c, so an observer moving with F1 sees them length contracted to one fourth of their normal length in the x direction, and he sees their clocks running at one fourth normal speed.

Now, we must figure out how that time dilation affects the expansion of space in F1. We know that, in F0, space expands at the rate of H

_{0} ≈ 2.5 x 10^-18/s in all directions. In F1, my first thought is that the galaxies are aging four times more slowly, so the space between galaxies must be expanding at .625 x 10^-18/s in all directions, according to F1 clocks.

This is where I run into doubts. Does this mean that F1 space expands at H

_{0} /4? Is it the same in all directions? I need some help, here, guys.

Looking ahead, I plan to set my F0 clocks to zero at the instant when the primordial plasma first became transparent. I place my F0 origin at Earth. Then I plan to synchronize my F1 clocks so that clocks at x = x' = 0 show the same time as Earth clocks; i.e. 13.7 billion years. I want to know what the CMBR would look like to an observer at the origin of F1 as he passes Earth.

On second thought, maybe I should set my F1 clocks to zero at the instant when the plasma became transparent at the origin of F1; i.e. the point in F1 which is passing Earth, now. Note that the plasma did not become transparent everywhere at once in F1. That's because events at different x coordinates which are simultaneous in F0 cannot also be simultaneous in F1. That's just basic special relativity.