Aliya Merali (Undergraduate, The College of New Jersey)

Can we Design a Simple Dark Energy

Engine?

Aliya Merali (Undergraduate, The College of New Jersey)

Robert Nemiroff (Michigan Technological University)

ThulsiWickramasinghe (The College of New Jersey)

Since ancient times, earth’s gravity has been used to harness

energy, converting gravitational potential energy kinetic

energy using systems such as the water wheel

Comparatively, an attempt to harness dark energy in a simple Comparatively, an attempt to harness dark energy in a simple engine of two tethered balls is conducted. The concept is similar to that of a waterfall converting potential energy to kinetic energy

Two identical balls falls in the earth’s gravitational field with

an initial separation. This separation will increase as a

function of time, and a tether will turn a motor.

As the ball falls the height changes as

`2cosh ⊕⊕ +

−=

Rt

gRxx o `

For 2

cosh2

⊕

⊕

⊕ +

−= tR

xx oo

⊕<< Rx

2

2

2)( tx

R

R

gxtx oo

−−≈ ⊕

⊕

As the two balls fall, is continuously increasingx∆

The concept is approached in a cosmological setting where the distances are small (We are envisioning a photon-tractor)

Consider a photon falling from a distant galaxy

eλ

ett =Time of emission

B

ott =

oλ

Time of observation

A

Where is the proper distanceand

)( ttcd op −=

1<<<zpd

We use the following form for the FRW Metric

Where are comoving coordinates,

( )[ ]sin)( 222222222 ϕθθχχ ddSdtRdtcds ++−=

( ( ), , )r S χ θ ϕ=

∫=ot

ttR

dtc

)(χ

In general, we find that the cosmological fluid obeys

where

t

Ω+Ω+

Ω+

Ω= Λ 243

2

2

2

aaaH

a

a krmo

&

zR

Ra

o +==1

1

In a vacuum domination

tH

tH

o o

oo

ee

aa Λ

Λ

Ω

Ω

=

ΛΩ== oHa

aH

&

1≈ΩΛ

This can be related to the scale factor, R, as

Iff

tH

tH

o o

oo

oo

ee

RR

e

Λ

Λ

Λ

Ω

Ω

=

1≈ΩΛ

( )[ ]11−

Ω= −Ω

Λ

Λ ttH

o

oooe

H

cRχ

Thus

Thus the proper distance to this object which has a look back time of , where is , or)( 0 tt − χ)( 0tRd p =

]1[1 )( −Ω

= −Ω

Λ

Λ ttH

o

pooe

H

cd

1≈ΩΛ

In the limiting case for where the objects are close together,

As is expected.

ΩΛoH

0≈− tto

)( ttcd op −≈

For , the proper velocity of B is

)(11

ttH ooecv−ΩΛ

−=

o

p

pdt

ddv =

1≈ΩΛ

For all z where ,

)(

11

ttH

pooe

zcv

−ΩΛ

+

−=

)1ln(1

)( zH

tto

o +Ω

=−Λ

1≈ΩΛ

For , the proper velocity becomes

as expected.

The proper velocity can be written as

czvp =

1<<z

Where and

)( ttcHv oop −Ω= Λ

tto ≈ 1<<z

We then calculate the acceleration of B with respect to A where

, , and . tto ≈ 1≈ΩΛ

o

p

dt

dvf =

1<<<z

)(2 ttcHf oo −Ω≈ Λ

odt

Now consider two balls separating where B is experience a force.

B

tt =

for small z, where and

A

ott = mfF =

)(2 ttmcHF oo −Ω≈ Λ

1≈ΩΛtto ≈

The potential energy of the system is

Where

pddFd =− φ

)( ttcd op −≈

222 )(2

1ttHmc ooo −Ω−= Λφφ

And

Thus, the potential becomes

)( ttdcdd op −≈

When , or , A and B are together. The potential energy is then . As the balls separate, the potential decreases, which is the opposite of the gravitational potential.

0=pd tto =

oφ

∞→− )( tt

−∞→φAs

Or as

Or as the balls diverge to

∞→− )( ott

∞→t

∞

oφφ −

)( tto −)0,0(

)( tto −

Gravitational

Cosmological

Because the potential energy diverges to from , power should be generated.

odt

dP

φ=−

−∞ oφ

Infinite power can be generated as the proper distance between the spheres increases. That is, as increases.

22/332 )( ttHmcP oo −Ω= Λ

tto −

In an approximation where

where

sunMm ≡

)(2/332 ttHcMP oosun −Ω≈ Λ

1≈Ωwhere

Where , which is 1000 light seconds away, or 1000 x the distance to the moon. Thus, the power generated is very small.

1≈ΩΛ

310≈− tto

WP 5.2≈

rrc

rGM

g

cG

ˆˆ

4

2

22

Λ+−≈

Λ−≈∇ ρπφ

In order for us to do this, the dark energy should dominate the local gravity.

In the weak field limit, general relativity gives:

rrc

rr

GMg ˆ

3ˆ

2

Λ+−≈

Therefore, there must exist a point at which g=0, occurring at3/1

2*

3

Λ

=c

GMr

In geometrized units for the sun, this distance is

pcr 10310

1477352* ≈

×= −

Thus in this case the length of the tether (photon-tractor!) should be > 103 pc! It is not possible to achieve such a length.

In a cosmological setting, we can differentiate

At ,

This infers a coasting period in the history of the universe when

Ω+Ω+

Ω+

Ω== Λ 243

22

2

2

aaaHH

a

a Krmo

&

0=a&&*aa =

This infers a coasting period in the history of the universe when . Using WMAP data, this happens at

Thus the engine is huge here and it becomes impractical.

1/3

11 0.7

2

ma za

∗ ∗Λ ∗

Ω= ⇒ = − ≈ Ω

*aa ≈

Discussion

While conceptually the prospect of harnessing dark energy is appealing, the practical applications are not currently plausible.

In order to create the large power outputs desired, large masses must be considered and separated by very large distances

Discussion

This would require a very long tether to connect the object, the mass of which could dwarf that of the objects

Instead we envision a photon-tractor [beam] Instead we envision a photon-tractor [beam]

My point is to make a suggestion about a scenario in which this energy could be harnessed, though not practical presently