Can we Design a Simple Dark Energy Engine? · 2010. 4. 12. · energy, converting gravitational...
Transcript of Can we Design a Simple Dark Energy Engine? · 2010. 4. 12. · energy, converting gravitational...
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