Large Extra Dimension Planck Scale

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 () to the fundamental (4+n)-dimensional scale () through the extra-dimensional volume factor:

In this convention: .

Written in MathML:

Solving for R gives: .

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 . (This differs from the non-reduced Planck mass by a factor of .)
• 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 differently depending on how the higher-dimensional action is normalized. This page follows the widely used 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: .
2. Compute the dimensionless ratio inside the parentheses: .
3. Take the power (here square root) and divide by to get in GeV⁻¹, then multiply by to get meters.

Numerically, this lands in the neighborhood of (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 : 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 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 or , and whether the volume factor is written as or . Those choices can change the quoted radius by factors of and .