1 Introduction

Relativity has been one of the most successful theories of the last century and is a cornerstone of modern physics. This review focuses on the modern experimental tests of one of the fundamental symmetries of relativity, Lorentz invariance. Over the last decade there has been tremendous interest and progress in testing Lorentz invariance. This is largely motivated by two factors. First, there have been theoretical suggestions that Lorentz invariance may not be an exact symmetry at all energies. The possibility of four-dimensional Lorentz invariance violation has been investigated in different quantum gravity models (including string theory [185

, 107 ], warped brane worlds [70

], and loop quantum gravity [120

]), although no quantum gravity model predicts Lorentz violation conclusively. Other high energy models of spacetime structure, such as non-commutative field theory, do however explicitly contain Lorentz violation [98

]. High energy Lorentz violation can regularize field theories, another reason it may seem plausible. Even if broken at high energies, Lorentz symmetry can still be an attractive infrared fixed point, thereby yielding an approximately Lorentz invariant low energy world [79 ]. Other ideas such as emergent gauge bosons [54 , 189 , 161 , 80 ], varying moduli [93 ], axion-Wess-Zumino models [30 ], analogues of emergent gravity in condensed matter [40

, 238 ], ghost condensate [34

], space-time varying couplings [177 , 50 ], or varying speed of light cosmologies [219 , 209

] also incorporate Lorentz violation. The ultimate fate of Lorentz invariance is therefore an important theoretical question.

We shall primarily focus on quantum gravity induced Lorentz violation as the theoretical target for experimental tests. If Lorentz invariance is violated by quantum gravity, the natural scale one would expect it to be strongly violated at is the Planck energy of $19 ~~ 10 GeV$ . While perhaps theoretically interesting, the large energy gap between the Planck scale and the highest known energy particles, the trans-GZK cosmic rays of $1011 GeV$ (not to mention accelerator energies of $~ 1 TeV$ ), precludes any direct observation of Planck scale Lorentz violation.

Fortunately, it is very likely that strong Planck scale Lorentz violation yields a small amount of violation at much lower energies. If Lorentz invariance is violated at the Planck scale, there must be an interpolation to the low energy, (at least nearly) Lorentz invariant world we live in. Hence a small amount of Lorentz violation should be present at all energies. Advances in technology and observational techniques have dramatically increased the precision of experimental tests, to the level where they can be sensitive to small low energy residual effects of Planck scale Lorentz violation. These experimental advances are the second factor stimulating recent interest in testing Lorentz invariance. One should keep in mind that low energy experiments cannot directly tell us whether or not quantum gravity is Lorentz invariant. Rather, they can only determine if the “state” that we live in is Lorentz violating. For example, it is possible that quantum gravity might be Lorentz invariant but contains tensor fields that acquire a vacuum expectation value at low energies [185 ], thereby spontaneously breaking the symmetry. Experiments carried out at low energies would therefore see Lorentz violation, even though it is a good symmetry of the theory at the Planck scale. That said, any discovery of Lorentz violation would be an important signal of beyond standard model physics.

There are currently a number of different theoretical frameworks in which Lorentz symmetry might be modified, with a parameter space of possible modifications for each framework. Since many of the underlying ideas come from quantum gravity, which we know little about, the fate of Lorentz violation varies widely between frameworks. Most frameworks explicitly break Lorentz invariance, in that there is a preferred set of observers or background field other than the metric [90 , 34 ]. However others try to deform the Poincaré algebra, which would lead to modified transformations between frames but no preferred frame (for a review see [186 ]). These latter frameworks lead to only “apparent” low energy Lorentz violation. Even further complications arise as some frameworks violate other symmetries, such as CPT or translation invariance, in conjunction with Lorentz symmetry. The fundamental status of Lorentz symmetry, broken or deformed, as well as the additional symmetries makes a dramatic difference as to which experiments and observations are sensitive. Hence the primary purpose of this review is to delineate various frameworks for Lorentz violation and catalog which types of experiments are relevant for which framework. Theoretical issues relating to each framework are touched on rather briefly, but references to the relevant theoretical work are included.

Tests of Lorentz invariance span atomic physics, nuclear physics, high-energy physics, relativity, and astrophysics. Since researchers in so many disparate fields are involved, this review is geared towards the non-expert/advanced graduate level, with descriptions of both theoretical frameworks and experimental/observational approaches. Some other useful starting points on Lorentz violation are [23 , 276 , 174 , 155 ]. The structure of this review is as follows. An general overview of various issues relating to the interplay of theory with experiment is given in Section 2. The current theoretical frameworks for testing Lorentz invariance are given in Sections 3 and 4. A discussion of the various relevant results from earth based laboratory experiments, particle physics, and astrophysics is given in Sections 5 and 6. Limits from gravitational observations are in Section 7. Finally, the conclusions and prospects for future progress are in Section 8. Throughout this review $jab$ denotes the Minkowski $(+ - - - )$ metric. Greek indices will be used exclusively for spacetime indices whereas Roman indices will be used in various ways. Theorists’ units $h = c = 1$ are used throughout. $EPl$ denotes the (approximate) Planck energy of $1019 GeV$ .