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I have a spaceship that can get to speeds very close to the speed of light.

How can this ship measure how fast it is going near the speed of light?

I would think one would measure how fast other stars fly by but does this even work if time dilation makes everything faster outside of my frame?

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You can use the relativistic Doppler effect

The relativistic Doppler effect is the change in frequency (and wavelength) of light, caused by the relative motion of the source and the observer (as in the classical Doppler effect), when taking into account effects described by the special theory of relativity.

The relativistic Doppler effect is different from the non-relativistic Doppler effect as the equations include the time dilation effect of special relativity and do not involve the medium of propagation as a reference point. They describe the total difference in observed frequencies and possess the required Lorentz symmetry.

Before starting the journey, your ship needs to have a set of reference light sources, with known spectra. During the trip, by measuring those spectra, relative velocity can be calculated.

In the picture below, taken from the linked page, the top half shows what Doppler shift to expect according to the direction in which the observer is looking at the light source in the relativistic case, while the bottom one shows the Doppler shift in the non relativistic case.

Doppler shift vs viewing angle

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    $\begingroup$ Would measuring the doppler shift against the cosmic background radiation work (anywhere in the universe) ? $\endgroup$
    – Fels
    Commented Jul 4, 2019 at 13:32
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    $\begingroup$ @Fels, cosmological subtleties apart, for which I am not able to answer, I am not sure we know its spectrum with the due precision, while we know the emission lines of several species. $\endgroup$
    – L.Dutch ♦
    Commented Jul 4, 2019 at 13:48
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    $\begingroup$ I would think cosmic background radiation, being radiation, would shift like any other radiation. You could measure the doppler shift of external radiation against a light source fixed to your ship. May I propose a space mermaid holding up a lantern on the bow? $\endgroup$
    – Willk
    Commented Jul 4, 2019 at 14:42
  • $\begingroup$ There is a very slight temperature dipole measured in the CMBR, from which it was calculated that Earth moves at a certain speed relative to the Hubble frame. CMBR should do just fine for relativistic speeds. $\endgroup$
    – John Dvorak
    Commented Jul 5, 2019 at 2:51
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It is important to note that travelling very near the speed of light is no different from being at rest - it all depends on the chosen frame of reference. Travelling at .999 c in one frame of reference = being totally still in another.

You can only measure your speed relative to other objects in space. There is no absolute speed. If you are heading towards a star, you can measure your speed relative to that star by the blueshift of the star's spectrum (which has a special signature depending on spectral type). Similarly, if you are moving away from a star, you can measure the redshift of its spectrum.

If you aren't heading directly away from or towards a star, you can measure the redshift and blueshift of stars to the side and compensate for the angle.

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    $\begingroup$ This is false due to cosmic background radiation. $\endgroup$
    – TLW
    Commented Jul 5, 2019 at 1:54
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    $\begingroup$ @TLW: How does cosmic background radiation make this false? $\endgroup$
    – Klaus Æ. Mogensen
    Commented Jul 5, 2019 at 7:19
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    $\begingroup$ @TLW cosmic background radiation gives a convenient reference frame, but it is not "absolute rest". See physics.stackexchange.com/questions/25928/… $\endgroup$
    – MaudPieTheRocktorate
    Commented Jul 5, 2019 at 11:33
  • $\begingroup$ @KlausÆ.Mogensen - "travelling very near the speed of light is no different from being at rest" -> This is false in the universe due to CMB. The faster you travel relative to the CMB rest frame, the more CMB you intercept. $\endgroup$
    – TLW
    Commented Jul 6, 2019 at 1:35
  • $\begingroup$ @MaudPieTheRocktorate - I make no claims that it is absolute rest. However, it is a universally-defined reference frame in our universe. $\endgroup$
    – TLW
    Commented Jul 6, 2019 at 1:36

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