Exploring Gravity with Optical Clocks

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Scientific Motivation

Why testing General Relativity (GR) and Special Relativity (SR)?

The prominent role of SR as well as GR as pillars of our understanding of space and time has been motivating experimenters to test its foundations and predictions with ever increasing accuracy. Additional motivation for better tests of SR and GR originates from (i) the search for a quantum gravity theory, (ii) the wish to unify all forces of nature and (iii) the fact that today SR is underlying high precision metrology.

• Since quantum theory and GR, which are both assumed to be valid universally, are not compatible, it is necessary to invent a new theory, called quantum gravity, which resolves these incompatibilities and which will lead to a modification of GR, SR and/or quantum theory. For example, prominent approaches towards a quantum theory such as loop gravity and string theory, lead to modified Maxwell equations which are not necessarily Lorentz covariant. Violations of Lorentz invariance also arise within quantum gravity induced modifications of the standard model and in non-commutative field theories. Any experimental hint in that direction would be of invaluable help in the formulation of such a new theory.
• A unification of all interactions will contribute much to a deeper understanding of physics. Models of unification often result in the prediction of violations of the basic assumptions of GR. In most cases one expects a violation of the universality of free fall (UFF) and of Local Position Invariance (LPI).
• Since 1983 the definition of the meter is based on the velocity of light. Of course, all other derived units are influenced by this definition. The uniqueness of this definition rests on the constancy of the speed of light and the universality of the influence of the gravitational field (which is always present) on the ticking rate of clocks. It is clear that experiments should confirm these facts in order to extend the foundations in metrology in view of future progress.

Is gravity to be described by a metric theory?

Einstein's General Theory of Relativity is based on two principles: First, gravity can be described by means of a space-time metric, and second, this metric is determined by the masses in the universe by means of the Einstein equations. A metric theory is mathematically described by a Riemannian geometry.

Until now all experiments confirm that gravity can indeed be described by a metric theory. This follows from the experimental validity of the Einstein Equivalence Principle (EEP). The EEP consists of

• the Universality of Free fall (UFF): all structureless bodies fall along the same path in a gravitational field, independent of their composition;
• Local Lorentz Invariance (LLI): The outcome of a (small-scale) experiment does not depend on the orientation and the velocity of the (inertial) laboratory.

These three principles together imply that gravity can be described by means of a space-time metric. Together with Einstein's field equations which relate the matter distribution to the gravitational field, we arrive at Einstein's GR. Due to their prominent role in this scheme, it is important to improve experimental tests of LPI and UFF.

The UFF has been found to be valid down to the 10-12-level and will be tested by the future space mission MICROSCOPE at the 10-15-level. Proposed space missions GG and STEP aim at a test at the 10-17-level and 10-18-level, respectively.

LPI has been tested at the 10-7 level with co-located optical clocks, by making use of the elliptical orbit of the Earth which carries the clocks to different distances from the Sun in the course of a year. The reason for this comparatively weaker estimate is that the gravitational redshift scales with U/c2 where U is the Newtonian potential. Only relatively small changes in U can be accessed from Earth. In the future, space missions, either in Earth orbit or approaching closely the Sun, can provide an up to ten-thousand-fold improvement.

The gravitational redshift is a consequence of metric theories of gravity. It can be measured by comparing the frequencies of clocks operating at different distance from a massive body. It has been tested at the 10-4 level by the rocket experiment Gravity Probe A (1976), and will be tested more accurately by the ESA mission ACES in 2012. Further improvement is aimed for with the mission “Space Optical Clocks” on the ISS and with mission concepts as mentioned above for the LPI test.

LLI, the local validity of Special Relativity (SR), can also be based on three basic principles.
Following the kinematical framework worked out by Robertson and Mansouri/Sexl, SR can be inferred from the following three observations:

• Isotropy of velocity of light,
• Independence of the velocity of light from the velocity of the laboratory,
• Doppler effect with Lorentz factor.

The isotropy of the velocity of light and the independence of the velocity of light from the velocity of the laboratory have been tested precisely using optical and microwave resonators. The Lorentz factor leading to the relativistic Doppler effect or, equivalently, to time dilation, has been tested with currently best accuracy using laser spectroscopy of Lithium ions moving at high velocity. (Of course, Lorentz invariance not only has to do with properties of the velocity of light or photons: also other particles like electrons, neutrons and protons obey the laws of SR. Their behaviour in this respect is tested very precisely with Hughes-Drever-type experiments e.g. using nuclear spectroscopy.)

The importance of SR for physics cannot be overestimated: Our understanding of space and time is based on SR, all experiments and observations are well described and explained within the formalism of SR. All established physical theories (i.e. of all know forces of nature) include the formalism of SR. Therefore, if one experiment finds a violation of SR, then the whole conceptual framework of physics will be affected.