relativistic positioning and navigation angelo tartaglia relgrav
TRANSCRIPT
RELATIVISTIC POSITIONING AND NAVIGATION
Angelo TartagliaRELGRAV
How could the Enterprise starship find her way in the
universe?
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More practical…
How to use a pulsar to find Starbucks
Cosmic GPS would employ pulsing stars, not satellites, as
celestial beacons
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Coordinates and positioning
• Space-time is a 4-dimensional generally curved metric manifold
• Gaussian coordinates may be used to localize events.
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Emission coordinates
Light cone
Clocks
Signalstime
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Null or light coordinates
Cartesian grid
Light rays grid
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Null vectors and waves
n̂,1Tcos,cos,cos,1T
02
The wave vector:
is a null vector
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The null basis
dcba ,,,
a
b
time
space
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Positioning in space-time
a
b
time
space
event
r
dd
d
cc
c
bb
b
aa
a
TTTTr
light coordinates
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Wave fronts
dabcdabc
abc
d
hyperplane
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b
b
time
space
a
a
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Uncertainty volume
dcba V
abc
acd
Td
Tb
4 dcba TTTTcl
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Locally uniform motion
Ta
Tb
time
space
Proper time t
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Light coordinates of an event
d,c,b,ad,c,b,a Txn
integerFrom simple linear equations
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..........
tt
tt
1x,tt
tt
1x,1x,tt
x
tt
1x,tt
1x,tt
1x,0x
48
12
48
142d
37
12
37
132c2b
15
122a
48
141d
37
131c
26
121b1a
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Uncertainty depends on clock
tt
t
t1
4xn4i,i
21i,i
n4i,i
As big as allowed by the linearity of the worldline
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Accelerated motion
...tTa
21
tTu
x 2a
a
a
aa
Four-velocity Four-acceleration
tau
2t a
a
max Maximum integration time
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A gravitational field
The gravitational field shows up when:
2tt
4u
Gravitational potential
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Pulsars as clocks
RELGRAV
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Two options
• X-ray pulsars• Radio-pulsars
Our choice is radio-pulsars
• ~ 1800 “clocks”• “Fixed” positions in the sky• Very stable clocks• Periods ≥ 1 ms
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Parkes observatory (Australia)
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Four real pulsars
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Static observer
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Uncertainties
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Eppur si muove
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Extension to moving sources
The method can be extended to nearby moving sources as clocks on satellites or on celestial bodies of the Solar system, provided one has the time dependence of the direction cosines of the null basis vectors
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Conclusion
• The problem of obtaining the local coordinates from the arrival times of pulses from remote sources has been solved
• The method naturally includes all relativistic effects
• The method can be applied both to pulsars and to clocks onboard satellites or celestial bodies
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• ML. Ruggiero, E. Capolongo, A. Tartaglia, Pulsars as celestial beacons to detect the motion of the Earth, IJMPD, in stampa (2011). •A. Tartaglia, ML. Ruggiero, E. Capolongo A null frame for spacetime positioning by means of pulsating sources, Advances in Space Research, 47, 645-653, 2011.• A. Tartaglia , Emission Coordinates for the Navigation in Space, Acta Astronautica, 67, 539-545, 2010• D. Bini, A. Geralico, ML. Ruggiero, A. Tartaglia, Emission vs Fermi coordinates: applications to relativistic positioning systems, Classical and Quantum Gravity, 25, 1-11, 2008.• ML. Ruggiero, A. Tartaglia, Mapping Cartesian Coordinates into Emission Coordinates: some Toy Models, IJMPD, 17, 311-326, 2008.