The Borde–Guth–Vilenkin (BGV) theorem is a
theorem
In mathematics and formal logic, a theorem is a statement (logic), statement that has been Mathematical proof, proven, or can be proven. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to esta ...
in
physical cosmology
Physical cosmology is a branch of cosmology concerned with the study of cosmological models. A cosmological model, or simply cosmology, provides a description of the largest-scale structures and dynamics of the universe and allows study of fu ...
which deduces that any
universe
The universe is all of space and time and their contents. It comprises all of existence, any fundamental interaction, physical process and physical constant, and therefore all forms of matter and energy, and the structures they form, from s ...
that has, on average, been
expanding throughout its history cannot be infinite in the past but must have a past
spacetime
In physics, spacetime, also called the space-time continuum, is a mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum. Spacetime diagrams are useful in visualiz ...
boundary.
It is named after the authors Arvind Borde,
Alan Guth and
Alexander Vilenkin
Alexander Vilenkin (; ; born 13 May 1949) is the Leonard Jane Holmes Bernstein Professor of Evolutionary Science and Director of the Institute of Cosmology at Tufts University. A theoretical physicist who has been working in the field of cosmolo ...
, who developed its mathematical formulation in 2003.
The BGV theorem is also popular outside physics, especially in religious and philosophical debates.
Description
In
general relativity
General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
, the
geodesics
In geometry, a geodesic () is a curve representing in some sense the locally shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connec ...
represent the paths that
free-falling particles or objects follow in curved
spacetime
In physics, spacetime, also called the space-time continuum, is a mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum. Spacetime diagrams are useful in visualiz ...
. These paths are the equivalent of the shortest path (straight lines) between two points in
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
. In
cosmology
Cosmology () is a branch of physics and metaphysics dealing with the nature of the universe, the cosmos. The term ''cosmology'' was first used in English in 1656 in Thomas Blount's ''Glossographia'', with the meaning of "a speaking of the wo ...
, a spacetime is said to be geodesically complete if all its geodesics can be extended indefinitely without encountering any singularities or boundaries. On the contrary, a spacetime that is geodesically past-incomplete features geodesics that reach a boundary or a singularity within a finite amount of
proper time
In relativity, proper time (from Latin, meaning ''own time'') along a timelike world line is defined as the time as measured by a clock following that line. The proper time interval between two events on a world line is the change in proper time ...
into the past.
In this context, we can define the average expansion rate as
:
where ''t''
i is an initial time ( is the proper initial time), ''t''
f a final time ( is the proper final time), and ''H'' is the expansion parameter, also called the
Hubble parameter.
The BGV theorem states that for any spacetime where
:
,
then the spacetime is geodesically past-incomplete.
The theorem only applies to classical spacetime, but it does not assume any specific mass content of the universe and it does not require gravity to be described by
Einstein field equations
In the General relativity, general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of Matter#In general relativity and cosmology, matter within it. ...
.
Derivation
For FLRW metric
Here is an example of derivation of the BGV theorem for an expanding homogeneous isotropic flat universe (in units of speed of light ''c''=1).
Which is consistent with
ΛCDM model, the current model of cosmology. However, this derivation can be generalized to an arbitrary space-time with no appeal to homogeneity or isotropy.
The
Friedmann–Lemaître–Robertson–Walker metric
The Friedmann–Lemaître–Robertson–Walker metric (FLRW; ) is a metric that describes a homogeneous, isotropic, expanding (or otherwise, contracting) universe that is path-connected, but not necessarily simply connected. The general form o ...
is given by
:
,
where ''t'' is time, ''x
i'' (''i''=1,2,3) are the spatial coordinates and ''a''(''t'') is the
scale factor. Along a timeline geodesic , we can consider the universe to be filled with comoving particles. For an observer with proper time ''τ'' following the
world line
The world line (or worldline) of an object is the path that an object traces in 4-dimensional spacetime. It is an important concept of modern physics, and particularly theoretical physics.
The concept of a "world line" is distinguished from c ...
, has a
4-momentum , where
is the energy, ''m'' is the mass and ''p''=, p, the magnitude of the 3-momentum.
From the
geodesic equation of motion, it follows that
where ''p''
f is the final momentum at time ''t''
f. Thus
:
,
where
is the Hubble parameter, and
:
,
''γ'' being the
Lorentz factor
The Lorentz factor or Lorentz term (also known as the gamma factor) is a dimensionless quantity expressing how much the measurements of time, length, and other physical properties change for an object while it moves. The expression appears in sev ...
. For any non-comoving observer ''γ''>1 and ''F''(''γ'')>0.
Assuming
it is follows that
:
.
Thereby any non-comoving past-directed timelike geodesic satisfying the condition
, must have a finite proper length, and so must be past-incomplete.
Implications
Current astronomical observations, show that the universe is expanding, thus the BGV implies that there must be a boundary or singularity in the history of the universe. This singularity has often been associated to the
Big Bang
The Big Bang is a physical theory that describes how the universe expanded from an initial state of high density and temperature. Various cosmological models based on the Big Bang concept explain a broad range of phenomena, including th ...
. However the theorem does not tell if it is associated to any other event in the past. The theorem also does not allow to tell when the singularity takes place, or if it is a
gravitational singularity
A gravitational singularity, spacetime singularity, or simply singularity, is a theoretical condition in which gravity is predicted to be so intense that spacetime itself would break down catastrophically. As such, a singularity is by defini ...
or any other kind of boundary condition.
Some physical theories do not discard the possibility of a non-accelerated expansion before a certain moment in time. For example, the expansion rate could be different from
up to the period of
inflation
In economics, inflation is an increase in the average price of goods and services in terms of money. This increase is measured using a price index, typically a consumer price index (CPI). When the general price level rises, each unit of curre ...
.
Limitations and criticism
Alternative models, where the average expansion of the universe throughout its history does not hold, have been proposed under the notions of emergent spacetime,
eternal inflation
Eternal inflation is a hypothetical inflationary universe model, which is itself an outgrowth or extension of the Big Bang theory.
According to eternal inflation, the inflationary phase of the universe's expansion lasts forever throughout most ...
, and
cyclic model
A cyclic model (or oscillating model) is any of several cosmological models in which the universe follows infinite, or indefinite, self-sustaining cycles. For example, the oscillating universe theory briefly considered by Albert Einstein in 1930 ...
s. Vilenkin and Audrey Mithani have argued that none of these models escape the implications of the theorem.
In 2017, Vilenkin stated that he does not think there are any viable cosmological models that escape the scenario.
Sean M. Carroll argues that the theorem only applies to
classical spacetime, and may not hold under consideration of a complete theory of
quantum gravity
Quantum gravity (QG) is a field of theoretical physics that seeks to describe gravity according to the principles of quantum mechanics. It deals with environments in which neither gravitational nor quantum effects can be ignored, such as in the v ...
. He added that Alan Guth, one of the co-authors of the theorem, disagrees with Vilenkin and believes that the universe had no beginning. Vilenkin argues that the Carroll–Chen model constructed by Carroll and Jennie Chen, and supported by Guth, to elude the BGV theorem's conclusions persists to indicate a singularity in the history of the universe as it has a reversal of the
arrow of time
An arrow is a fin-stabilized projectile launched by a bow. A typical arrow usually consists of a long, stiff, straight shaft with a weighty (and usually sharp and pointed) arrowhead attached to the front end, multiple fin-like stabilizers ca ...
in the past.
Joseph E. Lesnefsky, Damien A. Easson and
Paul Davies
Paul Charles William Davies (born 22 April 1946) is an English physicist, writer and broadcaster, a professor in Arizona State University and director of BEYOND: Center for Fundamental Concepts in Science. He is affiliated with the Institute ...
constructed an uncountable infinite class of classical solutions which have
and are geodesically complete. The authors claim that the geodesic incompleteness of inflationary spacetime is still an open issue. In this study, the authors argue that the previous investigations often did not use mathematically precise formulations of the BGV theorem and thus reached incomplete conclusions. Furthermore, there are examples of past-infinite models solving the problem of unbounded entropy growth which are geodesically complete.
Use in theology
Vilenkin has also written about the religious significance of the BGV theorem. In October 2015, Vilenkin responded to arguments made by theist
William Lane Craig
William Lane Craig (; born August 23, 1949) is an American Analytic philosophy, analytic philosopher, Christian apologetics, Christian apologist, author, and theologian. He is a professor of philosophy at Houston Christian University and at the T ...
and the
New Atheism movement regarding the existence of God. Vilenkin stated "What causes the universe to pop out of nothing? No cause is needed."
Regarding the BGV theorem itself, Vilenkin told Craig: "I think you represented what I wrote about the BGV theorem in my papers and to you personally very accurately."
See also
*
Kalam cosmological argument
*
Gibbons–Hawking–York boundary term
*
Gibbons–Hawking effect
*
Penrose–Hawking singularity theorems
References
Further reading
*
{{DEFAULTSORT:Borde-Guth-Vilenkin theorem
Eponymous theorems of physics
Physical cosmology