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# Fundamental Physics

Applications | |
---|---|

Title | Fundamental Physics |

Author(s) | ESA |

Edited by | GMV |

Level | Intermediate |

Year of Publication | 2013 |

GNSS may be considered as one of the first practical applications where relativistic effects are taken into account, not just from the theoretical point of view, but as a regular engineering constraint on the overall design requirements^{[1]}.

## Contents

# GNSS and Relativistic Mechanics

Why does GNSS need to take relativity corrections into account? The so-called Allan deviation for the best orbiting high-performance Cesium clocks in the GPS system is 4 nanoseconds per day. This means that after one day initializing a Cesium clock, its offset from a stable reference will deviate from its predicted value by not more than +/-4 ns with 67% probability and by not more than +/-8 ns with 95% probability. The effects arising from special and general relativity – gravitational blue shift, time dilation, and Sagnac effect – account for a total onboard clock advance of about 39 microseconds per day with respect to a clock located on the Earth’s ground laboratory, with sub-daily quasi-periodic variations of order 100 ns. The effect of the relativistic rate is 4 orders of magnitude above the Allan deviation but could eventually be merged with the determination of the natural rate of the clock. Nevertheless, without taking into account the various relativistic effects, the system would yield unacceptably large errors in positioning and in time transfer. For this reason, in GPS an adequate corresponding offset on the onboard clock frequency is imposed, while time-varying effects are corrected at the user level. In GLONASS and Galileo this correction is done at the user level^{[2]}^{[3]}^{[4]}^{[5]}^{[6]}.

## Synchronisation of Distant Clocks

Theoretically, as well as experimentally, it is known that absolute synchronisation of distant clocks is impossible, a direct implication of the finiteness of the speed of light. This might seem contradictory with respect to the need of current systems to define the constellation’s reference time through a synchronisation procedure. This apparent contradiction is generated by the misuse of general terms like synchronisation: In reality clocks are synchronised using a standard convention, called coordinate time synchronisation, involving the components of the space-time metric. Since one wishes to evaluate the ground to orbit clock synchronisation, it is advantageous to apply this convention in the context of the geocentric frame, which is defined by all clocks located on Earth’s geoid and at rest with respect to each other. Thanks to this convention, one can define the advance or delay of distant clocks with respect to each other. This is crucial not only for positioning itself, but also for applications related with time distribution. Coordinate time synchronisation is transitive, so that if clock A is synchronized with clocks B and C, then B and C will be synchronised with respect to each other. In the context of GNSS, clocks are located where the gravitational potentials differ and corrections typically of the order of 10 nanoseconds must be introduced.

## Relativistic Effects

Other relativistic effects must be taken into account when comparing the frequencies of distant clocks. Let us consider the case of a GNSS satellite and an observer on ground, both equipped with similar clocks. The satellite emits a signal with frequency f, measured with its on board clock. Although the observer on the ground takes into account the classical Doppler Effect, a different frequency will be measured. This discrepancy results from the gravitational blue shift and the second order Doppler Effect. These two effects are different for the satellite and for the observer moving with the Earth’s rotation. The overall effect results in a relative frequency excess of typically 4.7×10^{-10}, which as stated above corresponds to an onboard clock advance of 41 microseconds per day. Another relevant phenomena is the so-called Sagnac effect which concerns the propagation of electromagnetic signals in rotating reference frames. For the case of the GNSS, the Sagnac effect can amount to about 100 nanoseconds, corresponding approximately to 30 meters. Thus, satellite positioning systems provides the means to test the current theory of relativity. GPS has already been used to test the isotropy of the speed of light with a great precision and with the expected clocks improvement in the GNSS System it is expected to improve the test of the violation of the Local Positioning Invariance (LPI).

## Relativistic Effects and GNSS Evolution

The effects of general relativity are today taken into account by GNSS systems as corrections over a non-relativistic description of a “flat” space-time. This is sufficient for the current objectives of GNSS Systems but there are numerous reasons for looking beyond this approach, and explore the possibility of developing a fully general-relativistic conceptual framework for navigation. Among these reasons are:

- The possible advantages of a fully autonomous orbiting navigation system, that does not need to rely on Earth stations;
- The long-term perspective of solar-system and deep-space navigation;
- The theoretical value of developing a conceptual framework for navigation fully consistent with our fundamental understanding of space and time:

Ideally, one would like to develop a conceptual framework for navigation that could work in an arbitrary time-dependent spacetime and an arbitrary gravitational field. Theoretically, four clocks broadcasting their respective proper time define a coordinate system in spacetime. Such “GNSS-coordinates”, or “emission coordinates” are generically sufficient for defining a complete reference system in an arbitrary time dependent spacetime^{[7]}^{[8]}^{[9]}^{[10]}. Research on these relativistic and auto-located positioning constellations is ongoing, and may provide important inputs for the future of navigation. In the future, one may hope to integrate Earth-based, autonomous, and -say- pulsar based reference systems, to develop a truly global conceptual setting for general-relativistic navigation. Furthermore, research on the setting of a truly relativistic and self-referring positioning constellation, which requires at least four clocks broadcasting their respective proper times, would allow to downsize the ground segment as well as the total amount of satellites forming the constellation needed to achieve the same level of performance as systems based on a traditional architecture^{[9]}^{[10]}.

# Quantum Mechanics and GNSS

## Quantum communication

Satellites of the GNSS could host in their payloads hardware for experiments of space-to-space and ground-to-space links. These links could open new perspectives in quantum encoding techniques for secure data encryption and for tests of fundamental physics. The encoding of data on single photons is obtained by using the usual quantum communication schemes, namely phase encoding and interferometric techniques. These communication schemes have a higher capability than the classical equivalent ones, as each received photon can code many bits. Quantum encoding can also involve the superposition of quantum states and entanglement with strong non-local correlations that are the basis for the technique of quantum teleportation. In principle, entanglement would allow for a dramatic improvement of the time distribution among different ground-based stations, up to the picosecond level. ESA has already initiated several activities, at study level, in the domain of quantum communications. As a result, the Space QUEST experiment (Quantum Entanglement for Space ExperimenTs), to be flown on the ISS^{[11]}, may pave the way towards distribution of entangled photon pairs from orbit. Space-QUEST might turn out to be an important step towards the use of quantum entanglement as a new resource available to users of the GNSS.

## Relativity and Earth Gravitational Field

Relativistic effects, like the gravitational redshift, can be used to measure the Earth gravitational field by comparing orbital clocks relative to clocks located on the geoid. For instance, two clocks located in the Earth gravitational field in two different points differing by an altitude of 1 meter will have frequencies differing in relative value 10^{-16}; a space-clock located at an altitude of 400 km will be shifted with respect to the ground clock by 4.5×10^{-11}. Clock comparisons using the Atomic Clock Ensemble in Space (ACES), which is scheduled to fly onboard the ISS, linked through state of the art microwave capabilities will allow to measure differences in the Earth gravitational field corresponding to altitude differences of the order of 10 cm. This would open the door to relativistic geodesy. Since time distribution and experiments of Quantum Mechanics do have to take into account relativistic effects, one can envisage that if in the future the GNSS constellations will be equipped with clocks accurate enough to allow measuring the Earth gravitational field with a similar accuracy, a breakthrough in geodesy can be achieved, and a possible new range of services and space applications.
Understanding the local and global structure of space-time is one of the main tasks of modern physics. Progress in this domain is a major challenge in fundamental physics, as it can provide insights about the “new physics” resulting from the unification of the laws of nature — in particular, between the two pillars of current physics, namely general and quantum field theory. Advances in space physics allow going beyond the traditional role of observer of our immediate environment. Space is becoming a laboratory for new experiments and new technologies, which with unprecedented precision explore space-time itself.
Ideas for the evolution of GNSS, based on the improvement in current instruments, may allow for tests of fundamental physics in space concerning quantum technologies and general relativity.

# Credits

The information provided in this article has been compiled by GMV. In some cases, figures, tables and paragraphs have been extracted from the indicated references, in particular from the *Galileo Science Opportunity Document.*^{[12]}

# References

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- ^ Ashby, N., “Relativity in the Global Positioning System”, Living Rev. Relativity, 6(1), lrr-2003-1, (2003)
- ^ Rovelli, C., Phys. Rev. D 65 , 044017, 2002.
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- ^ Pascual-Sanchez, Ann. Phys. 16, 258, 2007.
- ^ Páramos, J., Bertolami, O., “Galileo satellite constellation and extensions to General Relativity” in Proceeding of the 1st Colloquium on scientific and fundamental aspects of the Galileo programme, Toulouse, 1-4 October 2007 (http://arxiv.org/abs/0710.3880).
- ^ Rovelli, C., Phys. Rev. D 65 , 044017, 2002.
- ^ Bahder, T., Phys. Rev. D 68 063005, 2003.
- ^
^{a}^{b}Coll B and Tarantola A, Proceedings Journées Syst‘emes de Référence (St. Petersburg, 2003) (St. Petersburg, Russia: Institut of Applied Astronomy of the Russian Academy of Science) pp 333–4 (See also http://coll.cc), 2004. - ^
^{a}^{b}Coll, B., Pozo, J. M., “Relativistic Positioning Systems: The emission coordinates”, Class. Quant. Grav., 23, 7395, 2006 - ^ Perdigues-Armengol, J., et al., ESA Bull. “Leap ahead in space communications” 137, 2009.
- ^ Galileo Science Opportunity Document, http://egep.esa.int/egep_public/file/GSOD_v2_0.pdf