Large Extra Dimension Planck Scale for ADD Gravity

Introduction: ADD gravity and the compactification radius

In the Arkani-Hamed–Dimopoulos–Dvali (ADD) picture, gravity propagates in 4 + n spacetime dimensions while Standard Model fields stay confined to a 3+1 dimensional brane. The apparent weakness of ordinary gravity can then be understood as the effect of gravitational flux spreading through the volume of the compact extra dimensions.

This calculator connects the higher-dimensional fundamental Planck scale (often written M★ and entered here in TeV) to the common compactification radius R of n flat extra dimensions. It reports R in meters, and the result panel also shows a 1/R scale so you can judge the compactification size in energy terms.

Core ADD relation for an n-torus

For n equal extra dimensions compactified on an n-torus with a shared radius R, the convention used here relates the reduced 4D Planck mass () to the fundamental (4+n)-dimensional scale () through the volume of the extra-dimensional space:

In this convention: .

Written in MathML:

Solving for R gives the expression used by the calculator: .

Units, constants, and ADD conventions

• The input M★ is entered in TeV because that is the natural scale for collider-style discussions; the calculator converts it to GeV internally 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 comes out in GeV⁻¹. The page converts that to meters with 1 GeV⁻¹ = 1.97327 × 10⁻¹⁶ m.
• Different ADD 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 compactification radius R

The value of R tells you how large each compact extra dimension must be 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.

At fixed M★, increasing n makes R drop quickly because the same gravitational scale is spread through more compact directions. Conversely, lowering M★ tends to increase R, sometimes dramatically.

Worked example: M★ = 10 TeV and n = 2 in the ADD formula

Here is the calculator's loaded ADD example, using M★ = 10 TeV and 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 the ADD radius changes

The table below highlights the main trend this calculator captures at fixed : increasing n reduces the radius required for a given fundamental scale.

Limitations and assumptions for the ADD estimate

This calculator is a compact ADD estimate, not a full phenomenology scan. It is most useful for seeing how M★ and n push the compactification radius up or down.

• Geometry: This calculator assumes n flat extra dimensions compactified on an n-torus with a single shared radius R. Realistic models can have unequal radii, curved spaces, warped geometries, or other compactification details.
• 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 for comparing ADD papers

If you are comparing this result to a paper, check whether the author uses or , and whether the compactification volume is written as or . Those choices can change the quoted radius by factors of and .

How to use this calculator for ADD compactification radii

1. Enter Fundamental scale M★ (TeV) as the positive TeV value you want to test.
2. Enter Number of extra dimensions n as a whole-number dimension count within the allowed range.
3. Click the button to compute the radius, then compare it with another M★ or n choice if you want to see how the ADD scale moves.

Formula: how the ADD radius estimate is built

The relation above is the algebraic core of the ADD calculator: once M★ and n are chosen, the tool solves for R using the reduced Planck mass and the n-dimensional volume factor. Because R appears inside an exponent, modest changes in either input can shift the compactification radius by orders of magnitude.

When you use the form, keep M★ in TeV and enter n as an integer extra-dimension count so the internal conversion stays consistent with the convention above.