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Paulo Luz
Newtonian Wormholes
VII Black Holes Workshop | 18 – 19 DecAveiro, Portugal
CENTRA
José P. S. Lemos and Paulo Luz, Gen. Relativ. Gravit. 46:1803 (2014); arXiv:1409.3231 [gr-qc]
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1. Introduction/Motivation
M. Abramowicz, G. Ellis, J. Horák, and M.Wielgus, “The perihelion of Mercury advanceand the light bending calculated in (enhanced)Newton's theory”, Gen. Relativ. Gravit. 46:1630 (2014).
Suggest an enhanced Newton’s theory ofgravitation, where gravity and thecurvature of space are not independent.
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1. Introduction
Enhanced Newton’s theory
(Poisson equation)
(Newton’s 2nd law)
(New equation)
(Euler equation)
(Continuity equation)
1. Introduction/Motivation
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1. Introduction1. Introduction/Motivation
Using this enhanced Newton’s theory,Abramowicz et al. computed the perihelion ofMercury advance and the light bending due toa central mass.
Their calculations agreed with thosemade in the framework of GeneralRelativity to accuracy of 𝒪 𝑀 𝑟 .
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2. Outline
Find a wormhole geometry using theEnhanced Newton’s theory framework.
Calculate the gravitational field andgravitational potential of the wormhole.
Compute the pressure of the fluid thatpermeates wormhole space.
Study the motion of a massive particle in thewormhole space.
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3. Construction of a Newtonian wormhole
We start with a spherically symmetric spacewith metric in the form:
Such that the Ricci scalar is
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3. Construction of a Newtonian wormhole
Impose a matter density
Using 𝑅 = 2𝑘𝜌
where 𝛽 ≡ 𝑏2𝑘𝛼.
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3. Construction of a Newtonian wormhole
Use an embedding diagram
The embedded surface will be axiallysymmetric, so 𝑧 = 𝑧 (𝑟), such that
Comparing both metrics
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3. Construction of a Newtonian wormhole
Impose the wormhole to be asymptotically flat
Verified for everyvalue of 𝐶1 and 𝛽.
Impose the throat condition
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3. Construction of a Newtonian wormhole
Substituting the value found for 𝐶1 on theembedding diagram equation we find:
Therefore, the throat condition also implies anupper bound for the parameter 𝛽 ≡ 𝑏2𝑘𝛼,
This restriction on the parameter 𝛽 also arisesfrom the flare out condition.
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3. Construction of a Newtonian wormhole
There is one more restriction in order to have awormhole geometry.
The proper radial distance, 𝑟∗, measuredby an observer from the throat, 𝑟 = 𝑏, to apoint of radial coordinate 𝑟 must be finite.
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3. Construction of a Newtonian wormhole
The fact that the proper radial distance mustbe well behaved implies:
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3. Construction of a Newtonian wormhole
Metric in the final form
with the shape function 𝑏 𝑟 given by
and the restriction on the parameter 𝛽 ≡ 𝑏2𝑘𝛼,
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3. Construction of a Newtonian wormhole
There’s a coordinate singularity at r=b.
Define a new coordinate 𝑙2 = 𝑟2 + 𝑏2
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4. Gravitational field and potential of the Newtonian wormhole
We have to solve the Poisson equation
With matter density
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4. Gravitational field and potential of the Newtonian wormhole
Given the symmetries of the system we used the Gauss law to find the gravitational field.
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4. Gravitational field and potential of the Newtonian wormhole
Gravitational field
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4. Gravitational field and potential of the Newtonian wormhole
Gravitational field behavior
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4. Gravitational field and potential of the Newtonian wormhole
Gravitational potential
Gravitational potential behavior
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5. Pressure support of the Newtonian wormhole
In the static case the Euler equation simplifies to
Imposing 𝑝 𝑙 → +∞ = 0 and numerically integrating the Euler equation we find
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6. Equations of motion of a test particle in the Newtonian Wormhole
We start with Newton’s 2nd law
The acceleration is given by
The particle’s path is described by a curve 𝑐such that
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6. Equations of motion of a test particle in the Newtonian Wormhole
The equations of motion are
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6. Equations of motion of a test particle in the Newtonian Wormhole
Pure circular motion: 𝑙 = 𝑙 = 0
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6. Equations of motion of a test particle in the Newtonian Wormhole
The general case can only be solved numerically.
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Thank you for your attention.