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In the book "The Elegant Universe" by Brian B. Greene, on chapter 6 it is stated that there's the so called "Planck tension" in string theory, and it is given a value of $10^{39}$ tons. This value is repeated some times.

But tension should be given in Newtons, not kilograms: is it saying $10^{42}$ kiloponds? so that'd be ~$10^{43}$ N ... But Planck force is ~$10^{44}$ N.

I think this is not a duplicate of the question What is tension in string theory?, or at least I need some clarification, because in that answer, it is stated:

"Because the string tension is not far from the Planck tension - one Planck energy per one Planck length or $10^{52}$ Newtons or so"

But Planck energy / Planck length = $1.9561·10^9$ J / $1.616199·10^{-35}$ m ~ $1.21·10^{44}$ N (Planck tension)

Or does $10^{52}$ N refers to string tension and not to Planck tension? But $10^{52}$ >> $10^{44}$ N, so they don't seem "not far" one from the other...

So being the disagreement between that answers's $10^{52}$ N, Greene's $10^{39}$ tons, and Planck tension $10^{44}$ N :

  • Which is a correct value in Newtons for string tension? It ranges then from $10^{43}$ N ($10^{39}$ tons) to $10^{52}$ N?

  • Is the value of Greene's book correct or is it an errata (this is unlikely, as I have seen this value in tons in comments from ths book) and if it is correct, how must that $10^{39}$ tons be converted to Newtons ? Is it correct $10^{43}$ N?

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circulosmeos
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1 Answers1

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  1. The Planck force is according to Wikipedia $$F_P~=~\frac{c^4}{G} ~\approx~ 1.21 \times 10^{44} N.$$

  2. Interestingly, the Planck force does not depend on the Planck constant $\hbar$. It is purely classical (as opposed to quantum mecanical)! In GR one often uses the reduced Planck force
    $$\frac{c^4}{8\pi G} ~\approx~ 4.81 \times 10^{42} N~\stackrel{g ~\approx~ 9.8~\frac{m}{s^2}}{\approx}~ 4.9 \times 10^{41}\text{ kg}~\approx~ 5.4 \times 10^{38}\text{ short ton},$$ in rough agreement with Brian Greene's estimate.

  3. The above is in 4 spacetime dimensions. The string tension $$T_0~=~\frac{1}{2\pi \hbar c \alpha^{\prime}},$$ lives in $$D~=~10$$ dimensions. The string tension $T_0$ depends on the model. But some general relationship with other physical constants can be inferred, see below.

  4. The string length is $$\ell_s~=~\hbar c \sqrt{ \alpha^{\prime} }.$$ For the remainder of this answer we work in unit where $c=1=\hbar$.

  5. The $D$-dimensional Newton's constant is given by $$G^{(D)}~=~ \left(\ell_P^{(D)}\right)^{D-2}~=~G V^{(D-4)}, $$ where $V^{(D-4)}$ is the unknown volume of an unknown compactified space. $\ell_P^{(D)}$ is the unknown $D$-dimensional Planck length.

  6. From the simplest string tree diagram, we expect $$G^{(D)}~\sim~ g^2 \ell_s^{D-2}, $$ where $g$ is the closed string coupling.

References:

  1. B. Zwiebach, A first course in String Theory, 2nd edition, 2009.
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