# Why don't strings have a Planck mass?

I understand that strings have a size of roughly the Planck length $$l_P$$ of $$10^{-35}$$ m.

If that is the case then one would expect that their mass would be roughly the Planck mass which is an enormous $$10^{19}$$ GeV.

(Strings that have small spins, like standard model particles, are about $$l_P$$ in length see https://physics.stackexchange.com/a/315166/22307)

In order to model the particles of the standard model their effective mass must be much smaller than the Planck mass.

Is the intrinsic Planck mass of a string largely cancelled by its negative gravitational binding energy so that its net mass is small?

For example assume that the gravitational binding energy of a string is roughly equal to its intrinsic mass energy then we have

$$\frac{GM^2}{R}\sim Mc^2$$

For a quantum object the uncertainty principle gives us the relationship

$$Mc\ R \sim \hbar$$

(I'm assuming a particle model such that its effective rest mass $$M$$ is entirely due to its internal momentum $$P=Mc$$ i.e. a zero rest mass particle confined to move around at the speed of light inside a box of size $$R$$)

Thus we find that

$$R \sim \sqrt{\frac{\hbar G}{c^3}} \sim l_P$$

Therefore, due to negative gravitational binding energy, Planck length strings are effectively massless. Thus they can reasonably model low-spin standard model particles which are very light compared to the Planck mass.

• The particles are not the string themselves, but their excitations. The latter need not have the same mass as the former -- in fact, the spectrum is gapless... – AccidentalFourierTransform Aug 11 '19 at 15:06
• But I would have thought that the ground state string still needs its mass "cloaked" somehow. – John Eastmond Aug 11 '19 at 15:12
• "I understand that strings have a size of roughly the Planck length" [citation needed] This sentence, and the rest of its reasoning, seem to be predicated on a thorough misunderstanding of string theory. As it stands, it is unanswerable because of that. – ACuriousMind Aug 11 '19 at 15:23
• Remotely relevant. But in comparison to string scales, all the particles we see are "basically" massless, or develop masses through the nonlinear interactions of massless gluons. – Cosmas Zachos Aug 11 '19 at 16:22