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]

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.

1. Introduction

Enhanced Newton’s theory

(Poisson equation)

(Newton’s 2nd law)

(New equation)

(Euler equation)

(Continuity equation)

1. Introduction/Motivation

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 .

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.

3. Construction of a Newtonian wormhole

We start with a spherically symmetric spacewith metric in the form:

Such that the Ricci scalar is

3. Construction of a Newtonian wormhole

Impose a matter density

Using 𝑅 = 2𝑘𝜌

where 𝛽 ≡ 𝑏2𝑘𝛼.

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

3. Construction of a Newtonian wormhole

Impose the wormhole to be asymptotically flat

Verified for everyvalue of 𝐶1 and 𝛽.

Impose the throat condition

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.

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.

3. Construction of a Newtonian wormhole

The fact that the proper radial distance mustbe well behaved implies:

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,

3. Construction of a Newtonian wormhole

There’s a coordinate singularity at r=b.

Define a new coordinate 𝑙2 = 𝑟2 + 𝑏2

4. Gravitational field and potential of the Newtonian wormhole

We have to solve the Poisson equation

With matter density

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.

4. Gravitational field and potential of the Newtonian wormhole

Gravitational field

4. Gravitational field and potential of the Newtonian wormhole

Gravitational field behavior

4. Gravitational field and potential of the Newtonian wormhole

Gravitational potential

Gravitational potential behavior

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

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

6. Equations of motion of a test particle in the Newtonian Wormhole

The equations of motion are

6. Equations of motion of a test particle in the Newtonian Wormhole

Pure circular motion: 𝑙 = 𝑙 = 0

6. Equations of motion of a test particle in the Newtonian Wormhole

The general case can only be solved numerically.

Thank you for your attention.