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Lisa Randall co-developed a geometric approach to the hierarchy problem with Raman Sundrum. Their Randall–Sundrum model uses a slice of five-dimensional Anti-de Sitter spacetime; its varying spacetime metric converts a fundamental large scale into a much smaller scale on the brane containing Standard Model fields. The mechanism relies on warping rather than an enormous compactification volume. It illustrates how assumptions about inaccessible spatial geometry can imply measurable particle excitations.^[ref]

The weak equivalence principle is universality of free fall for sufficiently small test bodies, independent of composition, when non-gravitational forces and self-gravity are negligible. Equality of inertial and passive gravitational mass makes this possible in the Newtonian limit. The Einstein equivalence principle adds local Lorentz invariance and local position invariance for nongravitational experiments: a small freely falling laboratory has the local physics of special relativity.

This motivates a universal spacetime metric. Freely falling unforced test particles follow geodesics, $u^{\nu }{\nabla }_{\nu }{u}^{\mu }=0$. At a point one can choose local inertial coordinates with vanishing connection, but curvature and tidal relative acceleration remain measurable over a finite region. Gravity cannot be transformed away globally. A uniformly accelerating small laboratory and a local gravitational field can have the same nongravitational experimental behaviour without having identical global geometry.

Clock comparisons predict a fractional rate difference of order $g\Delta h/c^2$ in a weak static field. The Pound-Rebka experiment tested the associated photon redshift through resonant gamma-ray emission and absorption at different heights, using controlled Doppler shifts to match the tiny energy displacement. Such a result tests the metric redshift prediction, not uniquely every component of the gravitational field equations.^[ref]

general relativity adds the dynamical Einstein field equations,The equivalence principle alone does not derive this unique equation; assumptions about covariance, dynamics and the Newtonian limit are also needed. The strong equivalence principle extends the universality claim to self-gravitating systems and gravitational experiments. Alternative metric theories can satisfy the weaker local principles while differing in gravitational dynamics, so composition tests, clock/redshift tests and self-gravity tests probe distinct claims.

Raman Sundrum and Lisa Randall proposed the Randall–Sundrum model to address the hierarchy problem. A warped spacetime metric produces an exponential relation between physical scales on two branes, even with only one compact additional spatial direction. In their 1999 construction, the Standard Model fields occupy one brane. The proposal makes the geometry of spacetime part of the explanation for a large separation of energy scales, with characteristic massive spin-two excitations as possible tests.^[ref]

Replacing the flat target metric in the Polyakov action by a spacetime metric $g_{\mu \nu }(X)$ gives the displayed string sigma model. The embedding fields $X^{\mu }$ map the worldsheet into target spacetime, while $h_{ab}$ is an independent worldsheet metric. Classical worldsheet diffeomorphisms and Weyl transformations permit conformal gauge; quantum consistency additionally requires vanishing sigma-model beta functions.