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Information about Comoving Coordinates

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In standard cosmology, 'comoving' distance or 'proper distance' is one of several distance measures used by cosmologists to define distances between objects.

Comoving coordinates

While general relativity allows one to formulate the laws of physics using arbitrary coordinates, some coordinate choices are natural choices which are easy to work with. Comoving coordinates are an example of such a natural coordinate choice. They assign constant spatial coordinate values to observers who perceive the universe as isotropic. Such observers are called "comoving" observers because they move along with the Hubble flow.

A comoving observer is the only observer that will perceive the universe, including the cosmic microwave background radiation, to be isotropic. Non-comoving observers will see regions of the sky systematically blue-shifted or red-shifted. Thus isotropy, particularly isotropy of the cosmic microwave background radiation, defines a special local frame of reference called the comoving frame. The velocity of an observer relative to the local comoving frame is called the peculiar velocity of the observer. Most large lumps of matter, such as galaxies, are nearly comoving, i.e., their peculiar velocities are low.

The elapsed time since the Big Bang according to a clock of a comoving observer is a measure of cosmological time. The comoving spatial coordinates tell us where an event occurs while cosmological time tells us when an event occurs. Together, they form a complete coordinate system, giving us both the location and time of an event.

Space in comoving coordinates is (on the average) static, as most bodies are comoving, and comoving bodies have static, unchanging comoving coordinates.

The expanding Universe has an increasing scale factor which explains how constant comoving coordinates are reconciled with distances that increase with time.

Comoving distance

Comoving distance is the distance between two points measured along a path defined at the present cosmological time. For objects moving with the Hubble flow, it is deemed to remain constant in time. The comoving distance from an observer to a distant object (e.g. galaxy) can be computed by the following formula:



where () is the scale factor.
is the time of emission of the photons detected by the observer
is the time "now".
Despite being an integral over time, this does give the distance that would be measured by a hypothetical tape measure at fixed time t. For a derivation see (Davis and Lineweaver, 2003) "standard relativistic definitions".

Symbols and substitute names for comoving distance
  • Some textbooks use the symbol .
  • proper distance is the name used by (Weinberg, 1972) for comoving distance.
  • Most textbooks and research papers define the comoving distance between comoving observers to be a fixed unchanging quantity independent of time, while calling the dynamic, changing distance between them 'proper distance'. On this usage, comoving and proper distances are numerically equal at the current age of the universe, but will differ in the past and in the future. (e.g. Davis and Lineweaver, 2003)

Uses of the comoving distance

Cosmological time is identical to locally measured time for an observer at a fixed comoving spatial position, that is, in the local comoving frame. Comoving distance is also equal to the locally measured distance in the comoving frame for nearby objects. To measure the comoving distance between two distant objects, one imagines that one has many comoving observers in a straight line between the two objects, so that all of the observers are close to each other, and form a chain between the two distant objects. All of these observers must have the same cosmological time. Each observer measures his distance to the nearest observer in the chain, and the length of the chain, the sum of distances between nearby observers, is the total comoving distance. It is important to the definition of comoving distance that all observers have the same cosmological age. For instance, if one measured the distance along a straight line or geodesic between the two points, one would not be correctly measuring comoving distance. Comoving distance is not quite the same concept of distance as the concept of distance in special relativity. This can be seen by considering the hypothetical case of a nearly empty universe, where both sorts of distance can be measured. In this thought experiment the value of comoving distance is not equal to the value of the distance as defined by special relativity. (Wright).

If one divides a comoving distance by the present cosmological time (the age of the universe) and calls this a "velocity", then the resulting "velocities" of "galaxies" near the particle horizon or further than the horizon can be above the speed of light. This apparent superluminal expansion is not in conflict with special or general relativity, and is a consequence of the particular definitions used in cosmology. Note that the cosmological definitions used to define the velocities of distant objects are coordinate dependent - there is no general coordinate independent definition of velocity between distant objects in general relativity (Baez and Bunn, 2006). The issue of how to best describe and popularize the apparent superluminal expansion of the universe has caused a minor amount of controversy. One viewpoint is presented in (Davis and Lineweaver, 2003).

Proper distance vs. comoving distance from small galaxies to galaxy clusters

Within small distances and short trips, the expansion of the universe during the trip can be ignored. This is because the travel time between any two points for a non-relativistic moving particle will just be the proper distance (i.e. the comoving distance measured using the scale factor of the universe at the time of the trip rather than the scale factor "now") between those points divided by the velocity of the particle. If the particle is moving at a relativistic velocity, the usual relativistic corrections for time dilation must be made.

See also

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Physical cosmology, as a branch of astronomy, is the study of the large-scale structure of the universe and is concerned with fundamental questions about its formation and evolution. Cosmology involves itself with studying the motions of the celestial bodies and the first cause.
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The age of the universe, in physics, is the time elapsed between the Big Bang and the present day. Current observations suggest that this is about 13.7 billion years, with an uncertainty of about +/-200 million years.
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Big Bang is the cosmological model of the universe whose primary assertion is that the universe has expanded into its current state from a primordial condition of enormous density and temperature.
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Blue Shift refers to a shortening of a transmitted signal's wavelength, and/or an increase in its frequency. The name comes from the fact that the shorter-wavelength end of the optical spectrum is the blue (or violet) end, hence, when visible light is compacted in
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cosmic microwave background radiation (most often abbreviated CMB but occasionally CMBR, CBR or MBR, also referred to as relic radiation) is a form of electromagnetic radiation discovered in 1965 that fills the entire universe [1].
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In physical cosmology, dark energy is a hypothetical form of energy that permeates all of space and tends to increase the rate of expansion of the universe. [1]
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In astrophysics and cosmology, dark matter is hypothetical matter of unknown composition that does not emit or reflect enough electromagnetic radiation to be observed directly, but whose presence can be inferred from gravitational effects on visible matter.
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Friedmann-Lemaître-Robertson-Walker (FLRW) metric is an exact solution of the Einstein field equations of general relativity; it describes a , isotropic expanding or contracting universe.
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The Friedmann equations are a set of equations in cosmology that govern the expansion of space in homogeneous and isotropic models of the universe within the context of general relativity.
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formation of galaxies is still one of the most active research areas in astrophysics; and, to some extent, this is also true for galaxy evolution. Some ideas, however, have gained wide acceptance.
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Hubble's law is a statement in physical cosmology which states that the redshift in light coming from distant galaxies is proportional to their distance. The law was first formulated by Edwin Hubble and Milton Humason in 1929[1] after nearly a decade of observations.
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In physical cosmology, cosmic inflation is the idea that the nascent universe passed through a phase of exponential expansion that was driven by a negative-pressure vacuum energy density.
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large-scale structure refers to the characterization of observable distributions of matter and light on the largest scales (typically on the order of billions of light-years).
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ΛCDM or Lambda-CDM is an abbreviation for Lambda-Cold Dark Matter. It is frequently referred to as the concordance model of big bang cosmology, since it attempts to explain cosmic microwave background observations, as well as large scale structure
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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.
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In physical cosmology, Big Bang nucleosynthesis (or primordial nucleosynthesis) refers to the production of nuclei other than those of H-1 (i.e. the normal, light isotope of hydrogen, whose nuclei consist of a single proton each) during the early phases of the
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This article or section may contain inappropriate or misinterpreted which do not the text.
Please help [ improve this article] by checking for inaccuracies. This article has been tagged since October 2007.
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redshift occurs when the electromagnetic radiation, usually visible light, that is emitted from or reflected off an object is shifted toward the (less energetic) red end of the electromagnetic spectrum.
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The shape of the Universe is an informal name for a subject of investigation within physical cosmology. Cosmologists and astronomers describe the geometry of the universe which includes both local geometry and global geometry.
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Structure formation refers to a fundamental problem in physical cosmology. The universe, as is now known from observations of the cosmic microwave background radiation, began in a hot, dense, nearly uniform state approximately 13.7 Gyr ago.
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Physical cosmology

  • Age of the universe
  • Big Bang
  • Blueshift
  • Comoving distance
  • Cosmic microwave background
  • Dark energy
  • Dark matter
  • FLRW metric
  • Friedmann equations
  • Galaxy formation
  • Hubble's law
  • Inflation

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For a timeline of the cosmos (or universe), see .


This timeline of cosmological theories and discoveries is a chronological catalog of the evolution of humankind's understanding of the cosmos over the last two-plus millennia.
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The ultimate fate of the universe is a topic in physical cosmology. Many possible fates are predicted by rival scientific theories, including futures of both finite and infinite duration.
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The Universe is defined as the summation of all particles and energy that exist and the space-time in which all events occur. Based on observations of the portion of the Universe that is observable, physicists attempt to describe the whole of space-time, including all matter and
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Astronomy is the scientific study of celestial objects (such as stars, planets, comets, and galaxies) and phenomena that originate outside the Earth's atmosphere (such as the cosmic background radiation).
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General relativity (GR) (aka general theory of relativity (GTR)) is the geometrical theory of gravitation published by Albert Einstein in 1915/16.[1] It unifies special relativity, Newton's law of universal gravitation, and the insight that gravitational
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Particle physics is a branch of physics that studies the elementary constituents of matter and radiation, and the interactions between them. It is also called "high energy physics"
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Quantum gravity is the field of theoretical physics attempting to unify quantum mechanics, which describes three of the fundamental forces of nature, with general relativity, the theory of the fourth fundamental force: gravity.
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Big Bang is the cosmological model of the universe whose primary assertion is that the universe has expanded into its current state from a primordial condition of enormous density and temperature.
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Distance measures are used in physical cosmology to give a natural notion of the distance between two objects or events in the universe. They are often used to tie some observable
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