Computation of Fermi and observational coordinates - UV · Fermi and observational coord. V. J. Bolós Deﬁnitions Applications Method 1: Taylor expansions Method 2: Iterative algorithm

Fermi andobservational

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V. J. Bolós

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Method 1:Taylorexpansions

Method 2:Iterativealgorithm

Examples

References

Spanish Relativity Meeting 2014, Valencia (Spain).

Computation of Fermi and observationalcoordinates

Vicente J. Bolós

Dpto. Matemáticas para la Economía y la Empresa,Universidad de Valencia.

September 3, 2014


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Fermi relative position

β observer with 4-velocity u at p.β′ test particle with 4-velocity u′s at qs.ψ spacelike geodesic orthogonal to u at p and withψ(0) = p, ψ(1) = qs.

=⇒ s = ψ(0).


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Observed relative position

β observer with 4-velocity u at p.β′ test particle with 4-velocity u′` at q`.λ past-pointing lightlike geodesic, λ(0) = p, λ(1) = q`.

w = λ(0) =⇒ sobs = projection of w onto u⊥.


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Fermi and Observational coordinates

Given a tetrad along β:

The Fermi coordinates of qs are the coordinates of s inthis tetrad.The observational coordinates of q` are the coordinates ofsobs in this tetrad.


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Distances:Fermi distance: ‖s‖Affine distance: ‖sobs‖

Relative velocities:Kinematic:

vkin :=1

−g(τψqspu′

s, u)τψqspu′

s − u

Spectroscopic:

vspec :=1

−g(τλq`pu′

`, u)τλq`pu′

` − u

Fermi:VFermi := ∇US + g (∇US,U) U

Astrometric:

Vast := ∇USobs + g (∇USobs,U) U


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Doppler effect:

ν ′

ν=

1√1− ‖vspec‖2

(1 + g

(vspec,

sobs‖sobs‖

))

Light aberration:

cos θ =cos θ′ − ‖vspec‖

1− ‖vspec‖ cos θ′

Gravitational lensing: There are different observed relativepositions sobs of a test particle at the same event p of theobserver.


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Method 1: Taylor expansions

Method 1

Find the local Taylor expansion for the transformation from apriori coordinates to Fermi / observational coordinates.

A priori =⇒ Fermi / observationalcoord. Taylor coord.

expansions

See e.g. D. Klein, P. Collas: General transformation formulas forFermi-Walker coordinates, Class. Quantum Grav. 25 (2008), 145019(arXiv: 0712.3838).


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Method 1: Taylor expansions

Some handicaps:

For increasing accuracy, we have to add terms of higherorder, and it could be very complex to achieve a highaccuracy for distant events.

There are problems in non-convex normal neighborhoods,where there is not uniqueness of Fermi / observationalcoordinates.


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Method 2: A numerical iterative algorithm

Method 2

Given an observer u at p and a test particle β′, we have to finds (Fermi coord.) and w (observational coord.).


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Spacelike case - Step 1

1. Choose an initial vector s0 ∈ u⊥ such that the geodesic ψ0starting from p with initial direction s0 is sufficiently closeto the test particle β′.

We need a concept of distance between curves: Working in acoord. system

{t, x1, x2, x3} with t = const a spacelike

foliation (whose leaves are regular submanifolds), the distancebetween two events p, q in the same leaf is given by

d(p, q) := ‖v‖,

such that there is a geodesic (in the leaf) from p to q withinitial tangent vector v .=⇒ Two curves c, c ′ are close if there exist p ∈ c and q ∈ c ′with the same coordinate time such that d(p, q) is small.


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Set n = 0.2. Find qgeo ∈ ψn and qpart ∈ β′ with the same coordinate

time that minimizes d(qgeo, qpart).


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3. Find qs by means of a Newton-Raphson method:We find qs.We redefine qpart as the event of β′ with the samecoordinate time as qs.Then, we find another qs, repeating this process until qsapproximates qs with the desired accuracy.

⇒ Quadratic convergence under some weak assumptions.


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Spacelike case - Steps 4-5

4. A linear approximation of s := exp−1p qs is given by

sνn+1 := sνn +(

J−1)νµ·(

q µs − qµgeo

),

where Jµν := ∂ν expµp s0 and sn is re-scaled in order to holdexpp sn = qgeo.

5. Redefine sn+1 as its projection onto u⊥:

sn+1 = sn+1 + g(u, sn+1)u.


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6. Set n = n + 1. Repeat the process (steps 2, 3, 4, 5) withthe geodesic ψn starting from p with initial direction sn,until we reach the desired accuracy.Otherwise, we should stop the algorithm if we arrive at apredetermined maximum number of iterations.

We can apply this algorithm again for another event β(t + ∆t)of the observer. The new initial vector s0 should be chosen asthe vector with the same coordinates as the corresponding finalvector sn computed for the previous event β(t).

=⇒ Choosing a sufficiently small time step ∆t assuresconvergence.


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Method 1 vs. Method 2

Both methods can be complementary. Moreover, Method 2overcomes the disadvantages of Method 1:

For increasing accuracy in Method 2, we have to makemore iterations.=⇒ This does not imply additional difficulty.

Method 2 is also valid in non-convex normalneighborhoods where there is not uniqueness of Fermi /observational coordinates: depending on the initial vectoryou can get different relative positions of the same testparticle at a given observer’s time.


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Example 1 - Schwarzschild (spherical coordinates,m = 1)

Stationary observer at r0 = 8, θ0 = π/2, ϕ0 = 0.Test particle with equatorial circular geodesic orbit withr1 = 4, θ1 = π/2, ϕ1 = π/2 at t = 0.


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Example 1 - Schwarzschild (spherical coordinates)

Spacelike case

n sn expp sn Rel. error0 (0, 0, 0, 1) (0, 10.65556, π/2, 0.82186) 2.11 (0,−5.92881, 0, 0.83524) (0, 5.98194, π/2, 1.42156) 5.9 · 10−1

2 (0,−7.22042, 0, 0.60405) (0, 4.05086, π/2, 1.56834) 1.4 · 10−2

3 (0,−7.24794, 0, 0.59718) (0, 4.00007, π/2, 1.57079) 2.1 · 10−5

4 (0,−7.24798, 0, 0.59717) (0, 4.00000, π/2, 1.57080) 6.2 · 10−8

Successive iterations of the algorithm for computing s. Theevent expp sn approximates the intersection event qs. Therelative error corresponds to the sum of the relative errors ofeach coordinate between expp sn and qs = (0, 4, π/2, π/2).


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Lightlike case

n wn expp wn Rel. error0 (−9.23760, 0, 0, 1) (−8.99173, 10.24564, π/2, 0.84457) 2.11 (−6.44519,−3.91665, 0, 0.40892) (−7.24116, 4.72029, π/2, 0.72302) 3.9 · 10−1

2 (−6.47318,−4.36531, 0, 0.30666) (−7.57135, 4.02002, π/2, 0.62785) 1.4 · 10−2

3 (−6.47266,−4.37700, 0, 0.30304) (−7.58081, 4.00001, π/2, 0.62321) 1.4 · 10−5

4 (−6.47266,−4.37701, 0, 0.30304) (−7.58082, 3.99999, π/2, 0.62321) 2.2 · 10−6

Successive iterations of the algorithm for computing w . Theevent expp wn approximates the intersection event q`. Therelative error corresponds to the sum of the relative errors ofeach coordinate between expp wn andq` = (−7.580825, 4, π/2, 0.623207).


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Example 2 - Kerr (Boyer-Lindquist coordinates,m = 1, a = 1/2)

Spacelike case

n sn expp sn Rel. error0 (−0.33333, 0, 0, 1) (−0.24490, 10.66478, π/2, 0.82236) 2.91 (−0.26677,−5.87287, 0, 0.80031) (−0.70619, 5.75206, π/2, 1.38644) 8.4 · 10−1

2 (−0.19748,−6.87825, 0, 0.59245) (−1.08418, 4.02663, π/2, 1.45063) 1.5 · 10−2

3 (−0.19628,−6.88956, 0, 0.58884) (−1.09232, 4.00019, π/2, 1.45005) 1.0 · 10−4

4 (−0.19627,−6.88964, 0, 0.58881) (−1.09238, 4.00000, π/2, 1.45005) 1.2 · 10−7

Successive iterations of the algorithm for computing s. Theevent expp sn approximates the intersection event qs. Therelative error corresponds to the sum of the relative errors ofeach coordinate between expp sn andqs = (−1.092382, 4, π/2, 1.450047).


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Lightlike case

n sn expp sn Rel. error0 (−9.59946, 0, 0, 1) (−9.28841, 10.14735, π/2, 0.85436) 2.01 (−6.80571,−4.00018, 0, 0.43270) (−7.94546, 4.55325, π/2, 0.74744) 2.9 · 10−1

2 (−6.76492,−4.33749, 0, 0.35627) (−8.17667, 4.00534, π/2, 0.66796) 3.4 · 10−3

3 (−6.76407,−4.34057, 0, 0.35537) (−8.17855, 3.99998, π/2, 0.66677) 4.4 · 10−6

Successive iterations of the algorithm for computing w . Theevent expp wn approximates the intersection event q`. Therelative error corresponds to the sum of the relative errors ofeach coordinate between expp wn andq` = (−8.178553, 4, π/2, 0.666767).


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Stationary observer at r0 = 8, θ0 = π/2, ϕ0 = 0.Test particle with equatorial circular geodesic orbit withr1 = 4, θ1 = π/2, ϕ1 = 0 at t = 0.


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r0 = 8


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r0 = 4


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r0 = 3


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Frequency shift:


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r0 = 8


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Examples 3-4

Frequency shift:


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References

Method 1:D. Klein, P. Collas: General transformation formulas forFermi-Walker coordinates, Class. Quantum Grav. 25(2008), 145019 (arXiv: 0712.3838).

Method 2:V. J. Bolos: An algorithm for computing geometricrelative velocities through Fermi and observationalcoordinates, Gen. Relativity Gravitation 46 (2014), 1623(arXiv: 1301.2932).

Thank you!

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Computation of Fermi and observational coordinates - UV · Fermi and observational coord. V. J. Bolós Deﬁnitions Applications Method 1: Taylor expansions Method 2: Iterative algorithm