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Metric expansion of space

2007 Schools Wikipedia Selection. Related subjects: Space (Astronomy)

          Physical cosmology

     * Age of the universe
     * Big Bang
     * Comoving distance
     * Cosmic microwave background
     * Dark energy
     * Dark matter
     * FLRW metric
     * Friedmann equations
     * Galaxy formation
     * Hubble's law
     * Inflation
     * Large-scale structure
     * Lambda-CDM model
     * Metric expansion of space
     * Nucleosynthesis
     * Observable universe
     * Redshift
     * Shape of the universe
     * Structure formation
     * Timeline of the Big Bang
     * Timeline of cosmology
     * Ultimate fate of the universe
     * Universe

            Related topics
     * Astrophysics
     * General relativity
     * Particle physics
     * Quantum gravity


   The metric expansion of space is a key part of science's current
   understanding of the universe, whereby spacetime itself is described by
   a metric which changes over time in such a way that the spatial
   dimensions appear to grow or stretch as the universe gets older. It
   explains how the universe expands in the Big Bang model, a feature of
   our universe supported by all cosmological experiments, astrophysics
   calculations, and measurements to date.

   The expansion of space is conceptually different from other kinds of
   expansions and explosions that are seen in nature. Our understanding of
   the "fabric of the universe" ( spacetime) requires that what we see
   normally as "space", "time", and " distance" are not absolutes, but are
   determined by a metric that can change. In the metric expansion of
   space, rather than objects in a fixed "space" moving apart into
   "emptiness", it is the space that contains the objects which is itself
   changing. It is as if without objects themselves moving, space is
   somehow "growing" in between them.

   Because it is the metric defining distance that is changing rather than
   objects moving in space, this expansion (and the resultant movement
   apart of objects) is not restricted by the speed of light upper bound
   that results from special relativity.

   Theory and observations suggest that very early in the history of the
   universe, there was an "inflationary" phase where this metric changed
   very rapidly, and that the remaining time-dependence of this metric is
   what we observe as the so-called Hubble expansion, the moving apart of
   all gravitationally unbound objects in the universe. The expanding
   universe is therefore a fundamental feature of the universe we
   inhabit—a universe fundamentally different from the static universe
   Albert Einstein first considered when he developed his gravitational
   theory.

Overview

   A metric defines how a distance can be measured between two nearby
   points in space, in terms of the coordinates of those points. A
   coordinate system locates points in an space (of whatever number of
   dimensions) by assigning unique numbers known as coordinates, to each
   point. The metric is then a formula which converts coordinates of two
   points into distances.

   For example, consider the surface of the Earth. This is a simple,
   familiar example of a non-Euclidean geometry. Because the surface of
   the Earth is two-dimensional, points on the surface of the earth can be
   specified by two coordinates - for example, the latitude, and
   longitude. Specification of a metric requires that one first specify
   the coordinates used. In our simple example of the surface of the
   Earth, we could chose any kind of coordinate system we wish, for
   example latitude and longitude, or X-Y-Z Cartesian coordinates. We then
   choose a metric which will be used to calculate the distance between
   nearby points in terms of the difference in the coordinates between
   them. In principle we can choose any one of a multitude of metrics, but
   in practice we choose a metric that seems to fit our purposes best. For
   example, local mapmaking often ignores curvature, whereas airline
   flights and distances are based upon a Great Circle. Since the Earth is
   not flat, the "distance" between two points (such as London and New
   York may differs depending on how it is to be measured (ie, the metric
   which is to be utilized). In measuring small distances on Earth, there
   is in theory always an effect due to curvature, but in practice the the
   Earth's curvature is so small as to be almost unnoticable for short
   distances.

   The metric of space appears from current observations to be Euclidean,
   on a large scale. The same cannot be said for the metric of space-time,
   however. On the surface of the Earth, we specified points by giving two
   coordinates. Because space-time is four dimensional, we must specify
   points in space-time by giving four coordinates. The most convenient
   coordinates to use for cosmology are called comoving coordinates.
   Because space appears to be Euclidean, on a large scale, one can
   specify the spatial coordinates in terms of x,y, and z coordinates,
   though other choices such as spherical coordinates are also commonly
   used. The fourth required coordinate is time, which is specified in
   comoving coordinates as cosmological time. The non-Euclidean nature of
   space-time manifests itself by the fact that the distance between
   points with constant coordinates grows with time, rather than remaining
   constant.

   Technically, the metric expansion of space is a feature of many
   solutions to the Einstein field equations of general relativity. In
   particular, if the cosmological principle is assumed with a
   time-varying universe the simplest solution allows for the distances in
   space to change with an evolving scale factor. This theoretical
   explanation provides a clean explanation of the observed Hubble's law
   which indicates that galaxies that are more distant from us appear to
   be receding faster than galaxies that are closer to us. In spaces that
   expand, the metric changes with time in a way that causes distances to
   appear larger at later times, so in our Big Bang universe, we observe
   phenomena associated with metric expansion of space. If we lived in a
   space that contracted (a Big Crunch universe) we would observe
   phenomena associated with a metric contraction of space instead.

   The first general relativistic models predicted that a universe which
   was dynamical and contained ordinary gravitational matter would
   contract rather than expand. Einstein's first proposal for a solution
   to this problem involved adding a cosmological constant into his
   theories to balance out the contraction, in order to obtain a static
   universe solution. It wasn't until the observations of Edwin Hubble in
   1929 confirmed that distant galaxies were all apparently moving away
   from us that scientists accepted that the universe was expanding. Until
   the theoretical developments in the 1980s no one had an explanation for
   why this was the case, but with the development of models of cosmic
   inflation, the expansion of the universe became a general feature
   resulting from vacuum decay. Accordingly, the question "why is the
   universe expanding?" is now answered by understanding the details of
   the inflation decay process which occurred in the first 10^-32 seconds
   of the existence of our universe. It is suggested that in this time the
   metric grew exponentially, causing space to change from smaller than an
   atom to around 100 million light years across.
   The expansion of the universe proceeds in all directions as determined
   by the Hubble constant today. However, the Hubble constant can change
   in the past and in the future dependent on the observed value of
   density parameters (Ω). Before the discovery of dark energy, it was
   believed that the universe was matter dominated and so Ω on this graph
   corresponds to the ratio of the matter density to the critical density
   (Ωm).
   Enlarge
   The expansion of the universe proceeds in all directions as determined
   by the Hubble constant today. However, the Hubble constant can change
   in the past and in the future dependent on the observed value of
   density parameters (Ω). Before the discovery of dark energy, it was
   believed that the universe was matter dominated and so Ω on this graph
   corresponds to the ratio of the matter density to the critical density
   (Ω[m]).

Measuring distances

   In expanding space, distance is a dynamical quantity which changes with
   time. There are several different ways of defining distance in
   cosmology, known as distance measures, but the most common is comoving
   distance.

   The metric only defines the distance between nearby points. In order to
   define the distance between arbitrarily distant points, one must
   specify both the points and a specific curve connecting them. The
   distance between the points can then be found by finding the length of
   this connecting curve. Comoving distance defines this connecting curve
   to be a curve of constant cosmological time. Operationally, comoving
   distances cannot be directly measured by a single Earth-bound observer.
   To determine the distance of distant objects, astronomers generally
   measure luminosity of standard candles, or the redshift factor 'z' of
   distant galaxies, and then convert these measurements into distances
   based on some particular model of space-time, such as the Lambda-CDM
   model.

Observational evidence

   It was not until the year 2000 that scientists finally had all the
   pieces of direct observational evidence necessary to confirm the metric
   expansion of the universe. However, before this evidence was
   discovered, theoretical cosmologists considered the metric expansion of
   space to be a likely feature of the universe based on what they
   considered to be a small number of reasonable assumptions in modeling
   the universe. Chief among these were:
     * the Cosmological Principle which demands that the universe looks
       the same way in all directions ( isotropic) and has roughly the
       same smooth mixture of material ( homogeneous).
     * the Copernican Principle which demands that no place in the
       universe is preferred (that is, the universe has no "starting
       point").

   To varying degrees, observational cosmologists have discovered evidence
   supporting these assumptions in addition to direct observations of
   space expanding. Today, metric expansion of space is considered by
   cosmologists to be an observed feature on the basis that although we
   cannot see it directly, the properties of the universe which scientists
   have tested and which can be observed provide compelling confirmation.
   Sources of confirmation include:
   Edwin Hubble presented the first observational evidence of an expanding
   universe.
   Enlarge
   Edwin Hubble presented the first observational evidence of an expanding
   universe.
     * Edwin Hubble demonstrated that all galaxies and distant
       astronomical objects were moving away from us ("Hubble's law") as
       predicted by a universal expansion. Using the redshift of their
       electromagnetic spectra to determine the distance and speed of
       remote objects in space, he showed that all objects are moving away
       from us, and that their speed is proportional to their distance, a
       feature of metric expansion. Further studies have since shown the
       expansion to be extremely isotropic and homogenous, that is, it
       does not seem to have a special point as a "centre", but appears
       universal and independent of any fixed central point.
     * In studies of large-scale structure of the cosmos taken from
       redshift surveys a so-called " End of Greatness" was discovered at
       the largest scales of the universe. Until these scales were
       surveyed, the universe appeared "lumpy" with clumps of galaxy
       clusters and superclusters and filaments which were anything but
       isotropic and homogeneous. This lumpiness disappears into a smooth
       distribution of galaxies at the largest scales in much the same way
       a Jackson Pollock painting looks lumpy close-up, but more regular
       as a whole.
     * the isotropic distribution across the sky of distant gamma-ray
       bursts and supernovae is another confirmation of the Cosmological
       Principle.
     * The Copernican Principle was not truly tested on a cosmological
       scale until measurements of the effects of the cosmic microwave
       background radiation in the dynamics of distant astrophysical
       systems. As reported by a group of astronomers at the European
       Southern Observatory, the radiation that pervades the universe is
       demonstrably warmer at earlier times. Uniform cooling of the cosmic
       microwave background over billions of years is explainable only if
       the universe is experiencing a metric expansion.

   Taken together, the only theory which coherently explains these
   phenomena relies on space expanding through a change in metric.
   Interestingly, it was not until the discovery in the year 2000 of
   direct observational evidence for the changing temperature of the
   cosmic microwave background that more bizarre constructions could be
   ruled out. Until that time, it was based purely on an assumption that
   the universe did not behave as one with the Milky Way sitting at the
   middle of a fixed-metric with a universal explosion of galaxies in all
   directions (as seen in, for example, an early model proposed by Milne).

   Additionally, scientists are confident that the theories which rely on
   the metric expansion of space are correct because they have passed the
   rigorous standards of the scientific method. In particular, when
   physics calculations are performed based upon the current theories
   (including metric expansion), they appear to give results and
   predictions which, in general, agree extremely closely with both
   astrophysical and particle physics observations. The spatial and
   temporal universality of physical laws was until very recently taken as
   a fundamental philosophical assumption that is now tested to the
   observational limits of time and space. This evidence is taken very
   seriously because the level of detail and the sheer quantity of
   measurements which the theories predict can be shown to precisely and
   accurately match visible reality. The level of precision is difficult
   to quantify, but is on the order of the precision seen in the physical
   constants that govern the physics of the universe.

Model analogies

   Because metric expansion is not seen on the physical scale of humans,
   the concept may be difficult to grasp. Three analogies, the
   ant-on-a-balloon analogy, the expanding rubber sheet analogy, and the
   raisin bread analogy, have been developed to aid in conceptual
   understanding. Each analogy has its conceptual benefits and drawbacks.

Ant on a balloon model

   The ant on a balloon model is a two-dimensional analog for
   three-dimensional metric expansion. An ant is imagined to be
   constrained to move on the surface of a huge balloon which to the ant's
   understanding is the total extent of space (see article on Flatland for
   more consequences of a two-dimensional constraint). At an early stage
   of the balloon-universe, the ant measures distances between separate
   points on the balloon which serves as a standard by which the scale
   factor can be measured. The balloon is inflated some more, and then the
   distance between the same points is measured and determined to be
   larger by a proportional factor. The surface of the balloon still
   appears flat, and yet all the points have appeared to recede from the
   ant, indeed every point on the surface of the balloon is proportionally
   farther from the ant than earlier in the life of the balloon universe.
   This explains how an expanding universe can result in all points
   receding from each other simultaneously. No points are seen to get
   closer together.

   In the limit where the ant is tiny and the balloon is enormous, the ant
   also cannot detect any curvature associated with the geometry of the
   surface (which is roughly an elliptical geometry for the outside
   surface of a curving balloon). To the ant, the balloon appears to be a
   plane extending out in all directions. This mimics the so-called "
   flatness" seen in our own observable universe which appears even at the
   largest scale to follow the geometrical laws associated with flat
   geometry. Like the ant on an enormous balloon, while we may be unable
   to detect curvature, on larger, unobservable scales there may be
   residual curvature. The shape of the universe we observe is driven to
   be flat no matter what starting conditions the universe had by the same
   cosmic inflation which caused the universe to begin expanding in the
   first place.

   In the analogy, the two dimensions of the balloon do not expand "into"
   anything since the surface of the balloon admits infinite paths in all
   directions at all times. There is some possibility for confusion in
   this analogy since the balloon can be seen by an external observer to
   be expanding "into" the third dimension (in the radial direction), but
   this is not a feature of metric expansion, rather it is the result of
   the arbitrary choice of the balloon which happens to be a manifold
   embedded in a third dimension. This third dimension is not
   mathematically necessary for two-dimensional metric expansion to occur,
   and the ant that is confined to the surface of the balloon has no way
   of determining whether a third dimension exists or not. It may be
   useful to visualize a third dimension, but the fact of expansion does
   not theoretically require such a dimension to exist. This is why the
   question "what is the universe expanding into?" is poorly phrased.
   Metric expansion does not have to proceed "into" anything. The universe
   that we inhabit does expand and distances get larger, but that does not
   mean that there is a larger space into which it is expanding.

Expanding rubber sheet model

   Similar to the ant on a balloon model, the expanding rubber sheet
   universe (ERSU) is given as a model that represents the expansion by
   ignoring the third dimension. Instead of relying on a balloon expanding
   into three dimensions, the ERSU model describes an infinite rubber
   sheet that is stretched in both directions. Heavy objects placed on the
   sheet create dips and dents of local curvature in much the same way
   massive galaxies curve spacetime in the gravitational wells of our
   universe. These objects all appear to be receding from each other
   unless they get caught in each other's gravitational wells (a process
   called virialization). The infinite rubber sheet stays infinite and two
   dimensional, but distances between points on the sheet steadily
   increase with the expansion. This model has the advantage over the
   balloon model of a macroscopically two-dimensional flat geometry which
   corresponds well to the measured three-dimensional (lack of) curvature
   in our observable universe.

Raisin bread model

   Animation of an expanding raisin bread model. As the bread doubles in
   size, the distances between raisins also double.
   Enlarge
   Animation of an expanding raisin bread model. As the bread doubles in
   size, the distances between raisins also double.

   The raisin bread model imagines galaxies as raisins in a raisin bread
   dough that will "rise" or "expand" when cooked. As the expansion
   occurs, each of the raisins gets farther from each of the other raisins
   while the raisins themselves stay the same size. The dough between
   raisins in this model acts as the space between galaxies while the
   raisins as "bound objects" are not subject to the expansion. This model
   is useful for explaining how it is that a standard ruler can be
   determined for measuring the expansion. In an empty universe, space
   serves as the only ruler and as rulers expand with space, there would
   be no way to distinguish between an expanding universe and a static
   universe. Only in a universe where there are objects which are bound
   and do not expand so that the rulers are independent of the expansion
   can the metric expansion be measured.

   Like the ant on the balloon model, this model also suffers from the
   problem that the raisin bread is expanding into the pan. To make the
   analogy to the universe, it is necessary to imagine raisin bread that
   has no observable edge. Expansion would still occur, but the question
   "what is the raisin bread expanding into?" would be meaningless.
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