Oh great, I finally get a day at the Maple Pavilion and the parks people have contracted for a remodeling of the nearby restroom facilities. So not only do I have the noise of the work crew’s drills, tile cutters and saws, I have to use a Port-O-Let. Anyway, it is starting out to be a beautiful day at the pavilion otherwise, so I’ll just enjoy the breeze, salt marsh and pine wood forest surrounding the pavilion and tune out the work crew.
I have to address the usage of the term ‘Lorentz boost’ in the context that it is used in Mersini-Houghton’s paper. The word boost, boosts, or boosted appears 44 times in the paper so it is obviously an important concept. Since no one who doesn’t have an attitude has offered to help me understand it let me start by quoting from post #50
QW said:
what I understand is that Lorentz boosts tie the standard cosmology (maybe you call the consensus the Lambda-CDM concordance model) to the current views of the multiverse. The current consensus is an inflationary model which, in the alternatives discussed by Mersini, is one bubble in the multiverse. Lorentz boosts initiate the bubbles and each boost nucleates from the initial conditions which are discussed in the Eternal Inflation model. She explains that the current multiverse models differentiate the bubbles from the inflationary spacetime landscape by referring to them as ‘true vacuum’ of lower energy while the inflating background is called the ‘false vacuum’ and is of higher energy.
And from post #39
…
Back to my question from a few posts ago about how Mersini uses the concept of Lorentz boost and Lorentz invariance and covariance.
Her reference to Lorentz boosts corresponds to the event the represents the initial conditions of a new bubble universe. Correspondingly, I equate her use of the term Lorentz boost to the concept of multiple big bangs. Using that interpretation each Big Bang gives a Lorentz boost to the new bubble (or arena in my terminology) that establishes an event horizon actuated by the light speed acceleration of the new bubble (Big Bang).
From post #51
That doesn't in itself explain what a Lorentz boost is. It's a very simple concept which is rather straightforward to describe in layman's terms, and it doesn't require any knowledge of cosmology to explain, so I'm inviting you to take another shot at defining it. Your answer would be a means of gauging how well you can personally understand papers like this without digging into the math and technical background.
And from post #55
A Lorentz boost is the operation of converting the description of a system from one observer's coordinate frame to the frame of another observer moving at some velocity relative to the first observer. Boosting a system to some velocity literally means you take the description of the system in some initial "rest" frame and then calculate what that description would be if the system were moving with said velocity relative to the chosen initial frame. …
It is like pulling teeth to get people like CptBork to participate and I shouldn’t give them such a hard time when they do. However, Bork is very dismissive of the uneducated and unwashed and so he has the same forum persona as AN. Both respond to me with egotistical and haughty attitudes, throwing around ad homs like “idiot” and “crank”. That is the reason I give them a hard time when they grace my threads with their presence.
However, rarely if ever in my extensive wasted time on forums over the past 10 years have I failed to heed what those obnoxious and deceitful types have said. And in spite of the general impression that they and their plethora of lackeys over time like to paint, my understanding of cosmology continually grows. The assertion that I am not interested in the real science is foolish and I write off those who say that as fools.
Bork could simply have participated in thread by offering the definition of Lorentz boost as part of the discussion. But no, he was intent on advancing is agenda of painting uneducated people as too dumb to appreciate science.
I will say that using his begrudgingly and disparagingly coughed up definition of Lorentz boost it does aid in understand the term as used by Mersini-Houghton.
My initial vague understanding of the term Lorentz boost found me saying: I equate her use of the term Lorentz boost to the concept of multiple big bangs. Using that interpretation each Big Bang gives a Lorentz boost to the new bubble (or arena in my terminology) that establishes an event horizon actuated by the light speed acceleration of the new bubble (Big Bang).
As Bork points out, using his definition my usage is unclear and undefined. Read the following portions of Mersini’s paper with this definition in mind and you will get a much clearer understanding of the term defined as: Lorentz boost = A Lorentz boost is the operation of converting the description of a system from one observer's coordinate frame to the frame of another observer moving at some velocity relative to the first observer. Boosting a system to some velocity literally means you take the description of the system in some initial "rest" frame and then calculate what that description would be if the system were moving with said velocity relative to the chosen initial frame.
The following might not copy over to the forum format very well but I think it will include every instance where boost was mentioned:
Abstract
1) Bubble universes nucleating
close to the initial conditions hypersurface have the largest Lorentz boosts and experience the
highest anisotropy. Consequently, their probability to collide upon formation is one.
Section 1
2) The location
dependent ’memory’ of the initial conditions and the
anisotropic distribution of bubbles [11, 15] is a function
of the boost factor of the observer with respect
to the initial conditions surface.
Section 2
3, 4, & 5) If an observer is positioned at some ξ = ξobs along Z
we can always bring this point to ξ′
obs = 0 by using the
Lorentz boost to transform to the observer’s frame.
The new coordinates in the boosted frame are
V ′ = γ (V − βZ) , Z′ = γ (Z − βV )
and, X′
i = Xi,W′ = W
where the relativistic boost factor γ = 1
(1−β2)1/2 is
γ = cosh(ξobs) and velocity β = tanh(ξobs).
6) The key point for our purposes, elaborated in Sec.3,
is that in its boosted frame the observer at ξ′
obs = 0
will see the initial conditions surface ai = a(ti) = 0
tilted to a new position a′
i = a′(ti) < 0 such that this
surface now cuts through and occupies portions of the
contracting phase [11].
7) The anisotropy per unit solid angle
depends on the position of the observer (θ′, ξobs) and
generally on the observer’s proper time τ from the initial
surface. Notice that the 4−volume diverges when
the boost γ becomes large.
Section 3
8-40) 3. Past and Future Incompleteness of Inflation
due to the Initial Conditions
An observer stationed at ξobs or a bubble nucleating
at that point, will experience uniform acceleration relative
to the preferred frame of the background. The
observer’s velocity relative to the preferred frame is
given by β = v/c with a Lorentz factor γ = 1/(1−β2)1/2 .
The observer’s proper time, τ, measured from the initial
conditions surface, is estimated from the Lorentz
factor γ [12] as
τ =
1
2
ln_γ + 1
γ − 1_ (8)
Observers have their velocities relative to the comoving
geodesic observer, redshifted by the scale factor
a(t) such that v ≃ 1
a(t) . In the far future (large
t, τ ) they align with the comoving observers of the
geodesics for t → +∞ since v → 0 . The Lorentz factor
with respect to the preferred frame can be found
from Eqns.(2, 4) through the expression, γ = dt
dτ ,
which for small τ → 0 becomes γ ≃ cosh(ξobs). From
Eqn.(8) it can be seen that in the limit of a large
proper time τ → +∞ from the I.C. surface, the
Lorentz factor tends to its minimum, γ → 1, because
in the limit τ → +∞, the velocity of bubbles nucleating
or observers located far from the initial conditions
surface relative to the preferred frame vanishes,
β = v
c → 0. From β2 = (1 − 1
γ2 ) ≃ 0 we thus have
γ ≃ 1.
The probability of collisions per unit time and unit
angle [11, 15] is proportional to the differential spacetime
4−volume, (Eqn.(6) )
λdV4
dτd′ =
λ
3
γ (1 + βcosθ′)) (9)
and it is clearly anisotropic. Here θ′ indicates the
direction of observation in the boosted coordinates.
The anisotropy towards the initial conditions surface
and in the observed distribution of bubble nucleations
and collisions [11, 15], depends on the location of the
observer, (ξobs, θ′, τ) since the location of the observer
determines the boost factor γ. The anisotropy disappears
only in the limit of large proper time τ → +∞,
since the vanishing velocities β ≃ 1
a(t) ≃ 0 there (discussed
above), align the boosted frame with the background
preferred frame.
We are interested to find out what happens to the
anisotropy in the limit of small proper time, τ ≃ ǫ ≪
1. At this stage, it important to notice that although
the boost factor is bound from below in the limit of
large proper times, (γ ≃ 1 and β ≃ 0 for τ ≃ ∞),
the boost γ becomes unbounded from above. With
this in mind, let us investigate the regime where the
boost, γ, may diverge: the regime of observers or
bubbles nucleating near the initial conditions surface,
τ ≪ 1. The initial conditions for eternal inflation at
t = ti = −∞ where the scale factor ai = a(ti) = 0,
make the assumption that no bubbles nucleate on the
surface ai = 0 or τ = 0. However bubbles can start
nucleating at some very small proper time, just ǫ ≪ 1
away from the initial conditions surface, τ = 0+ǫ ≪ 1.
Such initial conditions are artificial and lead to inconsistencies
of the theory as explained in Sec.4. The
key point here is that from Eqn.(8), observers stationed
only a small proper time from the initial conditions
surface, clearly have unbounded boost factors
γ → +∞ and large velocities β → 1, since in the limit
τ ≃ ǫ ≪ 1 the boost is given by
γ ≃ 1 +
1
ǫ
(10)
Thus γ → ∞ for ǫ → 0. Physically, the large velocity
and boost are due to the blueshifting effect from
geodesics convergence, as the initial conditions singularity
is approached (for τ → 0). For this reason,
a bubble nucleating, for example along Z, at
some small proper time τ = 0 + ǫ distance from the
initial conditions surface, will have a a very large
Lorentz boost γ ≃ cosh(ξobs) → +∞ and large velocity
β ≃ tanh(ξobs) → 1. A diverging relativis-
tic boost factor γ leads to a diverging 4−volume in
Eqn.(9), therefore to a probability one of bubble de-
struction from Eqn.(6) and the end of eternal inflation
.
This problem arises from the fact that observers located
near the initial conditions surface, with their
large relativistic boost factor γ, experience a highly
tilted initial conditions surface a′
i in their boosted
frame [11, 15]. In fact, their tilted I.C. surface a′
i
can be negative a′
i ≤ 0. The nearer the observer is to
the initial surface, then the larger their boost γ and
velocity β are. But, as shown in Eqn.(12) below, the
larger their boosts and velocities then the more negative
values their tilted I.C. surface a′
i ≤ 0 scans. Negative
values of a′ simply mean that, in the observer’s
boosted frame, the I.C. surface a′ cuts below the original
initial conditions ai = 0 boundary of eternal inflation
that separated inflating from contracting phases
of the global DeSitter geometry. Observers with the
tilted initial conditions surface, such that a′ < 0, thus
invade portions of the thermalized regions from the
contracting DS spacetime, which were ’forbidden’ by
the inflationary cutoff ai = 0. Since observers near
the initial conditions surface ai = 0 have larger boosts
γ then they cover larger volumes of the thermalized
spacetime region originally cut off from the inflationary
chart, than the faraway observers with vanishing
β’s and small boosts γ ≃ 1 . The relation between
the boost factor of the observer γ and the volume of
the noninflationary DeSitter region being scanned by
them, is problematic. This relation can be quantified
by recalling that the initial conditions surface at
˜ti = 0, ti = −∞, that separates the inflationary phase
from the contracting phase in the DS spacetime, is
given by the constraint V +W = ai. In the observer’s
boosted frame with coordinates
V ′ = γ [V − βZ] , Z′ = γ [Z − βV ] (11)
the initial conditions surface a′(ti) ’seen’ by the observer
becomes
a′ = V ′ +W′ = −βZ′ − γ−1 − 1_W′ ≤ 0 (12)
which is obtained from γ[V ′ + βZ′] = −W′. It
can be seen from Eqn.(12) that in its boosted frame,
the observer’s past light cone occupies a portion of
the contracting DS spacetime below the boundary for
eternal inflation ai = 0, (Fig.2.a), which was originally
cut off by the initial conditions boundary W = −V .
So, the larger the observer’s velocity β, the more tilted
a′
i becomes, implying that the more of the contracting
spacetime is invaded by the observer’s frame. But,
larger velocities relative to the preferred frame correspond
to observers and bubbles located near eternal
inflation’s surface of the initial conditions τ ≃ 0
from ai = 0. In short, the closer an observer is to
inflations’s initial conditions, the more of the ’forbidden’
spacetime region below the inflationary boundary
their chart occupies.
The tilting of the initial conditions surface and the
DS geometry as seen by the boosted frame are illustrated
in Fig.2. The hyperboloid of Fig.2.a shows the
global DS geometry obtained from Einstein equations.
The contracting and expanding phases are separated
at ˜ti = 0, ti = −∞ by the initial conditions surface
ai = eti = 0 indicated by the diagonal plane in Fig.2.a.
DS geometry as seen by the boosted observer is shown
in Fig.2.b. It can be seen that the initial conditions
plane a′
i ≤ 0 in the boosted frame is now tilted to
a new position. In Fig.2.c the DS spacetime of case
(b) seen by the observer in the boosted frame is superimposed
to the global DS geometry of Fig.2.a in
order to compare how much of the thermalized region
the boosted observer’s chart invades. The two global
phases in Fig.2.c are colored, red for the inflationary
half of the spacetime and blue for the contracting part.
For the case of a boost with β ≃ 0.9 depicted in Fig.2,
it can be seen that the tilted I.C. plane a′
i cuts below
the boundary of inflation ai = 0 and covers a large
part of the contracting spacetime (blue). The boosted
observer can thus come in contact with the ’forbidden’
(blue) thermalized regions of spacetime, initially cut
off from the inflationary chart via the I.C. boundary
ai = 0.
Why is the anisotropy towards the initial conditions,
experienced by the observers as a′
i ≤ 0, problematic
to the continuation of inflation? As we now
demonstrate, due to a′
i < 0, inflation can not be
future-eternal, instead it ends soon after the first bubbles
that form near the initial conditions surface. The
trouble comes from the fact that the volume of spacetime
below the global I.C. surface ai = 0, is completely
thermalized with no inflationary regions left since it
corresponds to the end of the contracting phase of
DeSitter (DS) geometry. Towards the end of the DS
contracting phase, (just below the ai = 0 boundary),
spacetime has contracted to its minimum size
near the boundary, all the bubbles have merged, have
grown to fill the whole spacetime, and thermalized.
From Eqn.(8) we know that bubbles forming near the
I.C. surface, with τ ≃ 0, have unbounded relativistic
boosts γ → +∞ and large velocities β = v
c → 1.
But in this limit γ → +∞, their 4− volume per unit
time and solid angle diverges when the boost is large
γ ≃ +∞ as follows from Eqn.(9). From Eqn.(5), a
diverging volume means that their probability to get
hit by the thermalized regions and other bubbles is
one. A diverging spacetime 4− volume of the highly
boosted observers γ ≫ 1, implies that the boosted
frame a′
i < 0 occupies too much of the contracting
DS phase in the global DS spacetime. Consequently,
the first bubbles that form near the I.C. surface soon
after the onset of infation, collide and are destroyed
immediately upon formation, with probability one.
All observers near the initial conditions region (with
proper time τ ≃ 0) have diverging boosts and velocities,
γ → +∞, v → c, as can be seen from Eqn.(10)
for the limit τ ≃ ǫ ≪ 1 in Eqn.(8). Therefore they
have highly tilted initial conditions surfaces a′
i < 0
allowing them to invade too much of the thermalized
region below the onset of inflation. According to
Eqn.(12) then, all bubbles near the onset of inflation
get hit with other bubbles upon formation and with
the thermalized regions originally not covered by the
eternal inflation spacetime, (regions below the initial
boundary ai = 0), resulting from Eqn.(9) and Eqn.(5).
Then inflation ends soon after the onset, and eternal
inflation becomes unlikely to be realized. As can be
seen, the problems stemming from the choice of initial
conditions in these scenarios are in close analogy with
the transplanckian problem of Hawking radiation in
which the frequency of the wavepackets is infinitely
blueshifted near the horizon.
Such an instability of the theory, the end of eternal
inflation, is a direct consequence of the choice of the
initial conditions, and it reflects the underlying nonlocal
relationship between the preferred frame (seen
as a′
i by the observer) and the inflationary metric,
Eqn.(12) with its initial conditions (fixed at ai = 0).
Section 4 Discussion
41 & 42)
Let us probe into the origin of these unexpected
difficulties in achieving future eternal inflation. Physically,
introducing a cutoff in the theory by imposing
the inflationary initial conditions at some special
time-slice, W = −V or equivalently ai = 0, leads to a
preferred frame that breaks Lorentz invariance. More
importantly the stationary (t = ti) preferred frame
of the inflationary background breaks the general covariance
of the theory [19], i.e. the consistency of the
Einstein Equations. As a result, observers with unbounded
Lorentz boosts γ positioned near the initial
conditions hypersurface, can scan portions of the ’forbidden’
contracting part of the DS spacetime below
the I.C. boundary W +V < 0. That part of the global
spacetime originally separated from the inflationary
region by imposing the initial conditions boundary, is
all thermalized. With probability one, bubbles near
the initial conditions region, with small proper distances
τ ≃ ǫ ≪ 1 thus large boosts γ ≃ 1
ǫ , Eqn.(10),
invade the thermalized regions originally exluded from
the inflationary spacetime via the boundary ai. Immediately
upon formation they get destroyed and inflation ends.
The root of the problem here lies in the
breaking of general covariance of this theory by the
I.C. of eternal inflation: near the I.C. surface, Einstein
Equations Gab = κTab are not satisfied since the
the divergence of the Einstein tensor and the stressenergy
tensor do not vanish, i.e. Bianchi identity is
not satisfied.
Figure 2 description
43 & 44)
IG. 2: (a) Global DS Spacetime with the Initial Condi-
tions surface ai separating the Inflating phase (upper half)
from the Contracting phase (bottom half). (b) The second
geometry indicates the global DS geometry in the boosted
frame of the observer. Notice the tilted initial conditions
surface a′
i the observer ’sees’. (c) The third DS spacetime
depicts the observers tilted boundary a′
i relative to ai. In-
flationary phase in (c) is in red and contracting phase in
blue and the 450 plane is ai = 0 separating the two phases.
We can see in (c) how much of the contracting (blue) DS
phase the observer’s boosted frame covers due to the ini-
tial boundary a′
i ≤ 0 piercing below the inflation’s initial
conditions ai = 0. The diagram is plotted for the repre-
sentative value _ ≃ 0.9.
[End of cut and paste]
Lol, it almost seems like I just pasted the whole paper but really, it is only the parts pertinent to the usage of "Lorentz boost".
I wanted to put it in the thread for reference because I often refer back to my threads to refresh my memory.
riginally Posted by Translation of quantum_wave