Nordtvedt effect

The Nordtvedt effect is a violation of the weak equivalence principle (WEP) that would cause bodies with different gravitational self-energy to fall at different rates in an external gravitational field. It is named after Kenneth Nordtvedt, who predicted it in 1968 within the framework of metric theories of gravity that do not satisfy the strong equivalence principle (SEP).¹ In general relativity, the SEP holds, so the effect is absent; but in scalar-tensor theories, such as Brans-Dicke theory, the gravitational binding energy contributes differently to inertial and gravitational mass, leading to a composition-dependent acceleration.

The most stringent test of the Nordtvedt effect comes from lunar laser ranging (LLR), which measures the Earth-Moon distance to millimeter accuracy. If the effect existed, the Moon would be displaced in its orbit by a few meters relative to the Earth's free-fall. Analysis of LLR data constrains the Nordtvedt parameter η to |η| < 1.3 × 10⁻¹³, consistent with zero.² This strongly supports general relativity and rules out many alternative theories.

The Nordtvedt effect is intimately connected to the Einstein equivalence principle (EEP) and the strong equivalence principle. The EEP is satisfied by all metric theories, but the SEP is only satisfied by general relativity. The presence of a Nordtvedt effect indicates a breakdown of the SEP, and its magnitude is proportional to the difference between gravitational and inertial mass, which in scalar-tensor theories is proportional to the scalar coupling strength.³ This makes the effect a powerful discriminator between general relativity and alternative theories.

The absence of the Nordtvedt effect in solar system experiments has profound implications. It constrains the parameters of scalar-tensor theories, such as the Brans-Dicke coupling constant ω, requiring ω > 40,000 from LLR.⁴ It also supports the idea that gravity is described by a metric theory with a purely tensor interaction, as in general relativity. However, the effect could still appear in strong-field regimes, such as in binary pulsars, where gravitational self-energy is large; current pulsar timing observations also show consistency with general relativity.⁵

Beyond the Earth-Moon system, the Nordtvedt effect has been studied in other contexts. For example, it would cause a differential acceleration of the Earth and Moon toward the Sun, which is precisely what LLR measures. In the early 1970s, there was a controversy over whether the effect existed in general relativity; Nordtvedt himself initially thought it might, but later calculations showed it cancels exactly.⁶ The effect is also relevant to the 'fifth force' hypothesis, which proposed a composition-dependent force; the absence of the Nordtvedt effect helped rule out many such proposals. Additionally, the effect has been considered in the context of modified gravity theories, such as f(R) gravity, where it can be suppressed by chameleon mechanisms.⁷

Equivalence principle

The principle that the gravitational and inertial masses are equal, leading to the universality of free fall.

Strong equivalence principle

The extension of the equivalence principle to include gravitational self-energy; it holds only in general relativity.

Scalar-tensor theory

A class of gravitational theories that include a scalar field in addition to the metric tensor, such as Brans-Dicke theory.

Lunar laser ranging

A technique that measures the distance from Earth to the Moon using laser pulses reflected by retroreflectors placed on the lunar surface.