Large Extra Dimension Planck Scale

Introduction: What this calculator does

In the Arkani-Hamed–Dimopoulos–Dvali (ADD) scenario, gravity propagates in 4 + n spacetime dimensions while Standard Model fields remain confined to a 3+1 dimensional brane. The observed (effective) weakness of gravity in four dimensions can then arise from the fact that gravitational flux “spreads out” into the extra-dimensional volume.

This calculator connects the higher-dimensional fundamental Planck scale (often written M★, entered here in TeV) to the common compactification radius R of n equal-size, flat, compact extra dimensions. It returns R in meters (and typically also millimeters if your results panel supports it).

Core relation (ADD with an n-torus)

For n extra dimensions compactified on an n-torus with common radius R, a commonly used convention relates the reduced 4D Planck mass (${\overline{M}}_{\text{Pl}}$) to the fundamental (4+n)-dimensional scale ($M_{\star }$) through the extra-dimensional volume factor:

In this convention: ${\overline{M}}_{\mathrm{Pl}}^2={M_{\star }}^{n+2}{\left(2\pi R\right)}^n$.

Written in MathML:

Solving for R gives: $R=\frac{1}{2\pi }{\frac{{\overline{M}}_{\text{Pl}}^2}{M_{★}^{n+2}}}^{\frac{1}{n}}$.

Units, constants, and conventions

• The input M★ is entered in TeV for convenience (collider-scale intuition). Internally it must be converted to GeV using 1 TeV = 10^3 GeV.
• The calculation uses the reduced Planck mass ${\bar{M}}_{\mathrm{P}\mathrm{l}}=2.435\times {10}^{18}\text{GeV}$. (This differs from the non-reduced Planck mass by a factor of $\sqrt{8\pi }$.)
• The intermediate result for R from the formula is in GeV⁻¹. Convert to meters using 1 GeV⁻¹ = 1.97327 × 10⁻¹⁶ m.
• Different papers sometimes place factors of $2\pi$ differently depending on how the higher-dimensional action is normalized. This page follows the widely used ${\left(2\pi R\right)}^n$ convention shown above.

Interpreting the radius R

The value of R tells you the approximate size of each compact extra dimension in this simplified ADD setup:

• Large R (e.g., microns to sub-millimeter): potentially testable via short-range gravity experiments that look for deviations from Newton’s inverse-square law at small distances.
• Intermediate/small R (e.g., nanometers and below): direct tabletop gravity tests become difficult; constraints typically come from astrophysics/cosmology and high-energy processes (model-dependent).
• Very tiny R: extra dimensions are effectively invisible at accessible distances/energies; the model resembles ordinary 4D gravity over macroscopic scales.

As you increase n at fixed M★, the required R generally decreases quickly. Conversely, lowering M★ (toward the TeV scale) tends to increase R.

Worked example

Example inputs: M★ = 10 TeV, n = 2.

1. Convert: $M_{★}=10\text{ TeV}={10}^4\text{ GeV}$.
2. Compute the dimensionless ratio inside the parentheses: ${\overline{M}}_{\mathrm{Pl}}^2/M_{\star }^{n+2}={(2.435\times {10}^{18})}^2/{\left({10}^4\right)}^4$.
3. Take the $\frac{1}{n}$ power (here square root) and divide by $2\pi$ to get $R$ in GeV⁻¹, then multiply by $1.97327\times {10}^{-16}$ to get meters.

Numerically, this lands in the neighborhood of $R\sim {10}^{-5}\text{ m}$ (tens of microns) for this specific convention—squarely in the regime where short-distance gravity tests are relevant. Your exact displayed value depends on rounding and the constants used.

Quick comparison table (how R changes)

The table below summarizes the qualitative trend at fixed ${\overline{M}}_{\text{Pl}}$: increasing n reduces the required radius for a given fundamental scale.

Limitations and assumptions

• Geometry: assumes n flat extra dimensions compactified on an n-torus with a single common radius R. Realistic models can have different radii, shapes, curvature, or warping.
• Convention dependence: the ${\left(2\pi R\right)}^n$ factor and the use of the reduced Planck mass are conventional choices. Other normalizations shift the numerical value of R by order-one factors.
• No phenomenology: this is an algebraic back-of-the-envelope mapping between scales; it does not compute collider cross sections, KK spectra details, astrophysical bounds, or cosmological constraints.
• Effective theory: treating gravity in 4+n dimensions with a sharp compactification radius is an effective description; UV completion details (string scale, strong coupling, brane effects) are not included.
• Input ranges: extremely small M★ or very large n can push the calculation into regimes where the simple ADD picture may be inconsistent with existing constraints or with the underlying assumptions.

Tip

If you are comparing to a paper, verify whether it uses $M_{\text{Pl}}$ or ${\overline{M}}_{\text{Pl}}$, and whether the volume factor is written as ${\left(2\pi R\right)}^n$ or $R^n$. Those choices can change the quoted radius by factors of $2\pi$ and $\sqrt{8\pi }$.

How to use this calculator

1. Enter Fundamental scale M★ (TeV) using the unit or time period shown by the field.
2. Enter Number of extra dimensions n using the unit or time period shown by the field.
3. Run the calculation and compare the output with a second scenario before acting on it.

Formula: how the estimate is built

The result can be read as result = f(a, b), where those inputs represent Fundamental scale M★ (TeV), Number of extra dimensions n. Keep money, time, distance, percentage, and count fields in the units requested by the form.