relativistic positioning and navigation angelo tartaglia relgrav
Post on 14-Jan-2016
213 Views
Preview:
TRANSCRIPT
RELATIVISTIC POSITIONING AND NAVIGATION
Angelo TartagliaRELGRAV
How could the Enterprise starship find her way in the
universe?
2011 March 03 RELGRAV A. Tartaglia 2
More practical…
How to use a pulsar to find Starbucks
Cosmic GPS would employ pulsing stars, not satellites, as
celestial beacons
2011 March 03 RELGRAV A. Tartaglia 3
2011 March 03 RELGRAV A. Tartaglia 4
Coordinates and positioning
• Space-time is a 4-dimensional generally curved metric manifold
• Gaussian coordinates may be used to localize events.
2011 March 03 RELGRAV A. Tartaglia 5
Emission coordinates
Light cone
Clocks
Signalstime
2011 March 03 RELGRAV A. Tartaglia 6
Null or light coordinates
Cartesian grid
Light rays grid
2011 March 03 RELGRAV A. Tartaglia 7
Null vectors and waves
n̂,1Tcos,cos,cos,1T
02
The wave vector:
is a null vector
2011 March 03 RELGRAV A. Tartaglia 8
The null basis
dcba ,,,
a
b
time
space
2011 March 03 RELGRAV A. Tartaglia 9
Positioning in space-time
a
b
time
space
event
r
dd
d
cc
c
bb
b
aa
a
TTTTr
light coordinates
2011 March 03 RELGRAV A. Tartaglia 10
Wave fronts
dabcdabc
abc
d
hyperplane
2011 March 03 RELGRAV A. Tartaglia 11
b
b
time
space
a
a
2011 March 03 RELGRAV A. Tartaglia 12
Uncertainty volume
dcba V
abc
acd
Td
Tb
4 dcba TTTTcl
2011 March 03 RELGRAV A. Tartaglia 13
Locally uniform motion
Ta
Tb
time
space
Proper time t
2011 March 03 RELGRAV A. Tartaglia 14
Light coordinates of an event
d,c,b,ad,c,b,a Txn
integerFrom simple linear equations
2011 March 03 RELGRAV A. Tartaglia 15
..........
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
2011 March 03 RELGRAV A. Tartaglia 16
Uncertainty depends on clock
tt
t
t1
4xn4i,i
21i,i
n4i,i
As big as allowed by the linearity of the worldline
2011 March 03 RELGRAV A. Tartaglia 17
Accelerated motion
...tTa
21
tTu
x 2a
a
a
aa
Four-velocity Four-acceleration
tau
2t a
a
max Maximum integration time
2011 March 03 RELGRAV A. Tartaglia 18
A gravitational field
The gravitational field shows up when:
2tt
4u
Gravitational potential
2011 March 03 A. Tartaglia 19
Pulsars as clocks
RELGRAV
2011 March 03 RELGRAV A. Tartaglia 20
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
2011 March 03 RELGRAV A. Tartaglia 21
Parkes observatory (Australia)
2011 March 03 RELGRAV A. Tartaglia 22
Four real pulsars
2011 March 03 RELGRAV A. Tartaglia 23
Static observer
2011 March 03 RELGRAV A. Tartaglia 24
Uncertainties
2011 March 03 RELGRAV A. Tartaglia 25
2011 March 03 RELGRAV A. Tartaglia 26
Eppur si muove
2011 March 03 RELGRAV A. Tartaglia 27
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
2011 March 03 RELGRAV A. Tartaglia 28
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
2011 March 03 RELGRAV A. Tartaglia 29
• 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.
top related