Local Lorentz and Local Position Invariance

Local Lorentz invariance and local position invariance test different directions in the space of possible experiments. The first asks whether a local nongravitational outcome changes when the freely falling apparatus is boosted or rotated. The second asks whether the same outcome changes when the apparatus is moved to another event or operated at another epoch. Both are needed, together with universal free fall, for the Einstein equivalence principle.

Required background. The comparisons use special-relativistic Lorentz transformations, local observer measurements, and the Newtonian potential when discussing clock comparisons in a weak field.

Helpful background. Tensor fields and tetrad projections provide coordinate-independent observables. Accelerated frames help separate a genuine orientation or position dependence from imperfect free fall.

Local Lorentz invariance

Take a nongravitational experiment inside a freely falling laboratory small enough that tidal corrections are negligible. Local Lorentz invariance (LLI) says that a dimensionless outcome does not depend on the constant velocity or spatial orientation of the apparatus. Two laboratories related by a local proper Lorentz transformation may decompose momenta and fields differently, but once the entire preparation and detector are transformed consistently, they infer the same local law.

Appropriate observables include a ratio of atomic transition frequencies, an angular distribution expressed relative to the apparatus tetrad, or a scattering cross section made dimensionless using a locally calibrated energy. A coordinate component such as $p^0$ is not by itself a suitable test because it changes under a boost even in exact special relativity. The invariant statement compares complete experimental arrangements and their tetrad readings.

Orientation tests and boost tests are two faces of LLI. A hypothetical transition frequency of the form

$$\frac{\nu}{\nu_0}=1+c_{ab}e_{(0)}^a e_{(0)}^b$$

would vary with the laboratory four-velocity if $c_{ab}$ contained a preferred-direction part. A term proportional to the metric alone would give $c_{ab}e_{(0)}^a e_{(0)}^b=\text{constant}$ and would not diagnose a violation. Coupling the spatial apparatus axes to a preferred tensor can instead create orientation dependence. Such dependence would violate LLI even if the background coefficients were identical at every event in the comparison model.

Will distinguishes these clock-anisotropy, preferred-frame, and boost comparisons from position-invariance tests Will 2014, § 2.1.2, pp. 9–12, PDF. The durable distinction is the class of variable being changed, not a particular experimental bound.

Local position invariance

Local position invariance (LPI) holds when the outcome of a local nongravitational experiment is independent of its spacetime location and epoch. “The same experiment” requires the same dimensionless preparation, the same local environmental variables, and the same orientation and velocity state. A change caused by temperature, electromagnetic contamination, or tidal gradients is not evidence of fundamental position dependence unless those influences have been controlled.

A simple hypothetical violation would be a dimensionless coupling that depends on a scalar background,

$$\alpha_{\mathrm{eff}}(x)=\alpha_0\bigl[1+\zeta\,\varphi(x)\bigr].$$

Atomic frequency ratios sensitive to $\alpha_{\mathrm{eff}}$ could then vary between events even though every local experiment remains Lorentz covariant. This violates LPI while preserving LLI. The dependence must be sought in dimensionless comparisons: variation of a dimensional constant alone can be shifted between unit conventions and is not an operational claim.

Gravitational redshift experiments are tests of LPI because metric gravity predicts a universal relation between clock rates and gravitational potential in the appropriate stationary weak-field regime. If different kinds of ideal clocks accumulated different fractional shifts between the same endpoints after nongravitational disturbances were removed, the discrepancy would signal nonuniversal local physics. The general classification and clock-comparison logic are summarized in Will 2014, § 2.1.3, pp. 12–15, PDF.

Independent logical possibilities

The two principles should not be merged into a generic claim that “physics is the same everywhere.” Consider three phenomenological ways of distinguishing their tests.

First, a constant preferred timelike vector can make outcomes depend on laboratory boost while introducing no explicit event dependence: LLI fails and LPI may hold. Second, dependence on the local value of a varying scalar coupling can change spectral ratios without selecting a local direction: LPI fails while that particular local comparison preserves LLI. Third, a composition-dependent gravitational response can be postulated independently of the clock and orientation protocols under study. In a complete theory these effects need not be independent: internal binding energies and conservation laws can relate free-fall and clock anomalies. That additional theoretical issue is explained in Will 2014, § 2.2.1, pp. 16–17, PDF.

Conversely, verifying all three within a domain supports EEP but does not establish the dynamics of gravity. A metric theory with additional gravitational fields can arrange universal matter coupling so that local nongravitational experiments obey LLI and LPI while its metric field equation differs from Einstein’s.

Locality and interpretation

Curvature produces legitimate position-dependent effects across an extended laboratory. If two ends of a large apparatus compare clocks, they are not conducting a single pointlike local experiment; the result contains propagation and tidal information. The local limit is controlled by quantities such as $|R|L^2$, where $L$ is the apparatus size. Shrinking $L$ separates an intrinsic violation from ordinary curvature variation.

The observer’s tetrad must also be specified at each event. Comparing differently oriented frames without accounting for their rotation can imitate an LLI signal, while comparing different gravitational environments without matching local conditions can imitate an LPI signal. The principles are therefore experimental statements about reproducible protocols, not assertions that coordinate components remain unchanged.

Check your understanding

Suppose a dimensionless spectral ratio depends on a smooth scalar field $\varphi(x)$ but is independent of the velocity and orientation of the freely falling apparatus at each event. Which component of EEP fails?

Solution

The failure is LPI: changing the event can change $\varphi$ and hence the spectral ratio. LLI remains intact because a scalar has the same value in all local Lorentz frames at that event and the protocol has no boost or orientation dependence. WEP is undecided until test-body trajectories are examined separately.

References

• Will, Clifford M. “The Confrontation between General Relativity and Experiment.” Living Reviews in Relativity 17, 4 (2014). DOI; arXiv.