Exploring Gravity Beyond Four Dimensions
One of the enduring puzzles in theoretical physics is the enormous disparity between the electroweak scale (around a few hundred GeV) and the Planck scale (~1019 GeV) at which gravity becomes comparable in strength to the other fundamental forces. The Arkani-Hamed–Dimopoulos–Dvali (ADD) model offers a radical proposal to address this hierarchy problem by postulating that our familiar four-dimensional spacetime is embedded in a higher-dimensional bulk with large extra dimensions. While Standard Model fields are confined to a (3+1)-dimensional brane, gravity can propagate into the additional spatial dimensions. As a result, the fundamental Planck scale in the higher-dimensional theory, denoted \(M_*\), could be much lower than the effective four-dimensional Planck scale \(M_{\text{Pl}}\), potentially near the TeV regime accessible to colliders.
The relationship between the observed Planck scale and the higher-dimensional fundamental scale depends on the volume of the extra dimensions. For simplicity, consider \(n\) extra dimensions compactified on an \(n\)-torus with common radius \(R\). The Gauss law for gravity in \(4+n\) dimensions implies the relation . Solving for the compactification radius yields . This calculator implements the formula using the reduced Planck mass \(M_{\text{Pl}} = 2.435\times10^{18}\,\text{GeV}\) and expresses \(R\) in meters.
The concept of large extra dimensions radically alters our understanding of gravity's apparent weakness. If \(M_*\) were near the TeV scale, gravitational interactions would become strong at much lower energies than previously thought, potentially allowing microscopic black holes to form in high-energy collisions or causing deviations from Newton's law at sub-millimeter distances. Experimental searches for such phenomena constrain the size of the extra dimensions, requiring \(R\) to be smaller than about 0.1 mm for \(n=2\), with even tighter bounds for higher \(n\). Nonetheless, the framework continues to inspire new avenues in model building, dark matter scenarios, and cosmology.
This calculator serves as an educational tool to explore how varying the number of extra dimensions and the fundamental scale affects the compactification radius. The input \(M_*\) is given in TeV to align with collider energy scales. Internally, the value is converted to GeV and substituted into the relation above. The result is presented in meters and millimeters, along with a qualitative classification of whether the radius falls within ranges probed by laboratory gravity experiments, collider signatures, or purely theoretical interest.
The calculations are purely algebraic: once the constants are set, the radius follows directly. We also compute the associated volume factor \((2\pi R)^n\) and the energy scale at which Kaluza–Klein excitations would appear, approximately \(1/R\). In experimental searches, Kaluza–Klein gravitons could lead to missing energy signals or modifications to cross sections.
The table below shows sample radii for different choices of \(n\) and \(M_*\), illustrating how rapidly the size of extra dimensions shrinks as either parameter increases:
| M★ (TeV) | n | R (m) | 1/R (GeV) |
|---|---|---|---|
| 1 | 2 | 3.8×10-3 | 5×10-4 |
| 5 | 3 | 1.5×10-9 | 1.3×1014 |
| 10 | 6 | 1.0×10-31 | 2×1020 |
As seen, pushing the fundamental scale higher or increasing the number of extra dimensions rapidly drives the compactification radius to microscopic values, well below current experimental reach. Yet even in such regimes, the framework can influence early-universe cosmology, gravitational wave production, or the propagation of high-energy cosmic rays.
To gain intuition, consider the case \(n=2\) and \(M_* = 1\,\text{TeV}\). The resulting radius \(R \approx 0.4\,\text{mm}\) lies just below current torsion-balance limits on deviations from Newtonian gravity. Thus, the absence of observed deviations places strong lower bounds on \(M_*\) for small \(n\). For \(n \geq 3\), the radius is much smaller, making direct tests more challenging. Instead, collider searches for missing energy or astrophysical observations of supernova cooling provide the leading constraints.
While large extra dimensions remain speculative, they exemplify the bold thinking that characterizes attempts to unify gravity with quantum mechanics. By adjusting the inputs in this calculator, one can rapidly assess the implications of different parameter choices and appreciate how geometry shapes the fundamental scales of nature.