  We study Kelvin modes and translational zero modes excited along a quantized
vortex and relativistic global string in superfluids and a relativistic field
theory, respectively, by constructing the low-energy effective theory of these
modes. We find that they become exact gapless Nambu-Goldstone modes only in a
system with infinite volume limit. On the other hand, in a system with the
finite volume, we find an imaginary massive gap causing the tachyonic
instability above some critical wavelength in the relativistic theory. We also
find in the non-relativistic theory that Kelvin modes with wavelengths longer
than some critical value propagate in the direction opposite to those with
shorter length, contrary to conventional understanding. The number of
Nambu-Goldstone modes also saturate the equality of the Nielsen-Chadha
inequality for both relativistic and non-relativistic theories.
