- Summary
- The intrinsic Planck mass of a string might be largely cancelled by its negative gravitational binding energy so that its net mass is small.

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.

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.