String Length, Coupling, and Newton Scale

String theory supplies several distinct length and coupling scales, and a black-hole calculation is reliable only when they are kept separate. This page derives the conversion among the string length, ten-dimensional Newton constant, compactification volume, and D-brane tension. The central lesson is simple but consequential: weak string coupling, small curvature in string units, and a large lower-dimensional Planck length hierarchy are independent conditions.

Required background. Gauge Theory and String-Theory Preparation defines open and closed strings, the dilaton, Chan–Paton factors, and the effective-action expansion used below.

Helpful background. Quantum Gravity places the Planck scale in a wider effective-field-theory setting; Quantum Information supplies later entropy language; and Black-Hole Geometry and Mechanics provides the horizon quantities whose normalization uses the Newton constant derived here.

Ten-dimensional normalization

Define the string length by $\ell_s^2=\alpha'$ and the asymptotic coupling by $g_s=e^{\Phi_\infty}$. In string frame the universal NS–NS action begins

$$S_{\mathrm{NS}}^{(s)} =\frac{1}{2\kappa_{10,0}^2} \int d^{10}x\,\sqrt{-g_s}\,e^{-2\Phi} \left(R_s+4(\nabla\Phi)^2-\frac{1}{12}H_3^2+\cdots\right),$$

where $2\kappa_{10,0}^2=(2\pi)^7\alpha'^4$. Expanding around $\Phi_\infty$ moves the factor $g_s^{-2}$ into the gravitational coupling,

$$2\kappa_{10}^2=(2\pi)^7g_s^2\alpha'^4, \qquad 16\pi G_{10}=2\kappa_{10}^2.$$

The overall action sign follows the site’s $(-,+,+,+)$ signature; what matters for the scale map is the common normalization of the Einstein term. The dimensions check immediately: $[G_{10}]=L^8$, exactly as required in ten dimensions. These relations, including the frame conversion and dimensional reduction conventions, are tabulated and derived in Peet 2000, §2.4, pp. 19–22.

The asymptotically normalized Einstein metric is

$$g^E_{MN}=e^{-(\Phi-\Phi_\infty)/2}g^s_{MN}.$$

Thus the two metrics agree at infinity, while a varying dilaton changes local proper lengths. A statement such as “the horizon is large in string units” must therefore name the frame and evaluate the relevant curvature invariant, rather than compare a coordinate radius with $\ell_s$.

Brane tension and charge

Write the D$p$-brane action in string frame as

$$S_{Dp}=-\mu_p\int_{\mathcal W_{p+1}}e^{-\Phi} \sqrt{-\det(P[g_s+B]+2\pi\alpha'F)} +\mu_p\int_{\mathcal W_{p+1}}P[C]\wedge e^{B+2\pi\alpha'F},$$

with

$$\mu_p=\frac{1}{(2\pi)^p\alpha'^{(p+1)/2}}, \qquad \tau_p^{(\infty)}=\frac{\mu_p}{g_s}.$$

Here $\mu_p$ is the DBI coefficient and R–R charge in this convention, whereas $\tau_p^{(\infty)}$ is the physical asymptotic tension after evaluating $e^{-\Phi_\infty}$. This distinction prevents a common double counting of $g_s^{-1}$. The scaling agrees with the disk origin of the brane action and with the D-brane identification in Polchinski 1995, Eqs. 8–10 and pp. 4725–4727.

Two useful checks follow. First, $\tau_p$ has dimension $L^{-(p+1)}$, so its worldvolume integral is dimensionless. Second, $\kappa_{10}^2\tau_p^2\sim \alpha'^{3-p}$ is independent of $g_s$: the closed-string exchange between two branes matches the one-loop open-string cylinder after the modular transformation.

Reduction to the lower-dimensional Newton scale

For an unwarped Einstein-frame product $M_d\times X_n$, with $d+n=10$ and physical compact volume

$$V_n=\int_{X_n}d^ny\,\sqrt{g_X},$$

substitution into the Einstein–Hilbert term gives

$$\frac{1}{G_d}=\frac{V_n}{G_{10}}, \qquad \ell_{P,d}^{\,d-2}=G_d=\frac{G_{10}}{V_n}.$$

On a rectangular torus, $V_n=(2\pi)^n\prod_iR_i$. Increasing the compact volume therefore weakens lower-dimensional gravity. In a warped compactification the correct coefficient is a warped Einstein-frame integral, not the unweighted coordinate volume; moduli-dependent Weyl rescaling is also required before identifying $G_d$.

For the D3-brane, flux quantization gives $L^4=4\pi g_sN\alpha'^2$. Supergravity curvature control asks for $L^2/\alpha'=(4\pi g_sN)^{1/2}\gg1$, while closed-string loops ask for the appropriate effective coupling to be small. Large $N$ can satisfy both in the familiar window, but neither condition follows from $g_s\ll1$ alone.

This example also distinguishes the ten-dimensional Planck scale from the string scale. From $G_{10}\sim g_s^2\ell_s^8$, one obtains $\ell_{P,10}\sim g_s^{1/4}\ell_s$. At weak coupling the Planck length is shorter than the string length, so massive string modes appear before ten-dimensional quantum-gravity loops as energy is raised. Compactification can reverse the apparent ordering in the lower-dimensional theory because the Planck scale also contains $V_n$. A reliable hierarchy is therefore stated in terms of dimensionless ratios such as $r_h/\ell_s$, $r_h/\ell_{P,d}$, and $r_h/R_i$ for the same physical solution. None may be inferred from another without the charge and volume scalings.

Reliability checks

A black-hole scale table should contain at least $\ell_s$, the local curvature radius $L_{\mathcal R}$, the ten- or lower-dimensional Planck length, Kaluza–Klein radii, $g_s$, and the charge-dependent effective coupling. The leading two-derivative description requires $\alpha'/L_{\mathcal R}^2\ll1$; perturbative string loops require a small local loop parameter; dimensional reduction requires retained energies below the first relevant Kaluza–Klein mass. These inequalities can fail independently.

As a normalization cross-check, compactifying all six extra directions at $R_i=\ell_s$ gives $V_6=(2\pi)^6\alpha'^3$ and hence

$$G_4=\frac{(2\pi)^7g_s^2\alpha'^4}{16\pi(2\pi)^6\alpha'^3} =\frac{g_s^2\alpha'}{8}.$$

Changing a radius changes $G_4$ even though $G_{10}$ is fixed; changing $g_s$ changes both the gravitational strength and physical D-brane tension. This is why a coupling variation at fixed four-dimensional Newton constant generally requires a compensating compactification change.

Exercises

Derive the four-dimensional Planck length for a rectangular six-torus and determine how all radii must scale with $g_s$ if $G_4$ is held fixed.

Solution

Using $V_6=(2\pi)^6\prod_{i=1}^6R_i$ and $G_4=G_{10}/V_6$ gives $G_4=g_s^2\alpha'^4/(8\prod_iR_i)$, which reduces to the normalization displayed above when every $R_i=\ell_s$. More generally, at fixed $\alpha'$ and fixed $G_4$, the product $\prod_iR_i$ must scale as $g_s^2$. If all radii scale together, each obeys $R_i\propto g_s^{1/3}$. The check is dimensional: the product has dimension $L^6$, while only its coupling dependence is being varied. This correlated scaling also warns that the Kaluza–Klein gap changes, so holding the Newton constant fixed does not hold the effective theory fixed.

References

• Peet, Amanda W. “TASI Lectures on Black Holes in String Theory.” 2000. arXiv:hep-th/0008241. Open manuscript.
• Polchinski, Joseph. “Dirichlet-Branes and Ramond–Ramond Charges.” Physical Review Letters 75 (1995): 4724–4727. DOI.