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Saturday, October 23, 2021

Stress–energy tensor

From Wikipedia, the free encyclopedia
 
Contravariant components of the stress–energy tensor.

The stress–energy tensor, sometimes called the stress–energy–momentum tensor or the energy–momentum tensor, is a tensor physical quantity that describes the density and flux of energy and momentum in spacetime, generalizing the stress tensor of Newtonian physics. It is an attribute of matter, radiation, and non-gravitational force fields. This density and flux of energy and momentum are the sources of the gravitational field in the Einstein field equations of general relativity, just as mass density is the source of such a field in Newtonian gravity.

Definition

The stress–energy tensor involves the use of superscripted variables (not exponents; see tensor index notation and Einstein summation notation). If Cartesian coordinates in SI units are used, then the components of the position four-vector are given by: x0 = t, x1 = x, x2 = y, and x3 = z, where t is time in seconds, and x, y, and z are distances in meters.

The stress–energy tensor is defined as the tensor Tαβ of order two that gives the flux of the αth component of the momentum vector across a surface with constant xβ coordinate. In the theory of relativity, this momentum vector is taken as the four-momentum. In general relativity, the stress–energy tensor is symmetric,

In some alternative theories like Einstein–Cartan theory, the stress–energy tensor may not be perfectly symmetric because of a nonzero spin tensor, which geometrically corresponds to a nonzero torsion tensor.

The components of the stress-energy tensor

Because the stress–energy tensor is of order 2, its components can be displayed in 4 × 4 matrix form:

In the following, k and range from 1 through 3:

  1. The time–time component is the density of relativistic mass, i.e., the energy density divided by the speed of light squared, while being in the co-moving frame of reference. It has a direct physical interpretation. In the case of a perfect fluid this component is

    where is the relativistic mass per unit volume, and for an electromagnetic field in otherwise empty space this component is

    where E and B are the electric and magnetic fields, respectively.
  2. The flux of relativistic mass across the xk surface is equivalent to the density of the kth component of linear momentum,
  3. The components
    represent flux of kth component of linear momentum across the x surface. In particular,
    (not summed) represents normal stress in the kth co-ordinate direction (k = 1, 2, 3), which is called "pressure" when it is the same in every direction, k. The remaining components
    represent shear stress (compare with the stress tensor).

In solid state physics and fluid mechanics, the stress tensor is defined to be the spatial components of the stress–energy tensor in the proper frame of reference. In other words, the stress energy tensor in engineering differs from the relativistic stress–energy tensor by a momentum-convective term.

Covariant and mixed forms

Most of this article works with the contravariant form, Tμν of the stress–energy tensor. However, it is often necessary to work with the covariant form,

or the mixed form,

or as a mixed tensor density

This article uses the spacelike sign convention (−+++) for the metric signature.

Conservation law

In special relativity

The stress–energy tensor is the conserved Noether current associated with spacetime translations.

The divergence of the non-gravitational stress–energy is zero. In other words, non-gravitational energy and momentum are conserved,

When gravity is negligible and using a Cartesian coordinate system for spacetime, this may be expressed in terms of partial derivatives as

The integral form of this is

where N is any compact four-dimensional region of spacetime; is its boundary, a three-dimensional hypersurface; and is an element of the boundary regarded as the outward pointing normal.

In flat spacetime and using Cartesian coordinates, if one combines this with the symmetry of the stress–energy tensor, one can show that angular momentum is also conserved:

In general relativity

When gravity is non-negligible or when using arbitrary coordinate systems, the divergence of the stress–energy still vanishes. But in this case, a coordinate-free definition of the divergence is used which incorporates the covariant derivative

where is the Christoffel symbol which is the gravitational force field.

Consequently, if is any Killing vector field, then the conservation law associated with the symmetry generated by the Killing vector field may be expressed as

The integral form of this is

In special relativity

In special relativity, the stress–energy tensor contains information about the energy and momentum densities of a given system, in addition to the momentum and energy flux densities.

Given a Lagrangian Density that is a function of a set of fields and their derivatives, but explicitly not of any of the spacetime coordinates, we can construct the tensor by looking at the total derivative with respect to one of the generalized coordinates of the system. So, with our condition

By using the chain rule, we then have

Written in useful shorthand,

Then, we can use the Euler–Lagrange Equation:

And then use the fact that partial derivatives commute so that we now have

We can recognize the right hand side as a product rule. Writing it as the derivative of a product of functions tells us that

Now, in flat space, one can write . Doing this and moving it to the other side of the equation tells us that

And upon regrouping terms,

This is to say that the divergence of the tensor in the brackets is 0. Indeed, with this, we define the stress–energy tensor:

By construction it has the property that

Note that this divergenceless property of this tensor is equivalent to four continuity equations. That is, fields have at least four sets of quantities that obey the continuity equation. As an example, it can be seen that is the energy density of the system and that it is thus possible to obtain the Hamiltonian density from the stress–energy tensor.

Indeed, since this is the case, observing that , we then have

We can then conclude that the terms of represent the energy flux density of the system.

Trace

Note that the trace of the stress–energy tensor is defined to be , where

When we use the formula for the stress–energy tensor found above,

Using the raising and lowering properties of the metric and that ,

Since ,

In general relativity

In general relativity, the symmetric stress–energy tensor acts as the source of spacetime curvature, and is the current density associated with gauge transformations of gravity which are general curvilinear coordinate transformations. (If there is torsion, then the tensor is no longer symmetric. This corresponds to the case with a nonzero spin tensor in Einstein–Cartan gravity theory.)

In general relativity, the partial derivatives used in special relativity are replaced by covariant derivatives. What this means is that the continuity equation no longer implies that the non-gravitational energy and momentum expressed by the tensor are absolutely conserved, i.e. the gravitational field can do work on matter and vice versa. In the classical limit of Newtonian gravity, this has a simple interpretation: kinetic energy is being exchanged with gravitational potential energy, which is not included in the tensor, and momentum is being transferred through the field to other bodies. In general relativity the Landau–Lifshitz pseudotensor is a unique way to define the gravitational field energy and momentum densities. Any such stress–energy pseudotensor can be made to vanish locally by a coordinate transformation.

In curved spacetime, the spacelike integral now depends on the spacelike slice, in general. There is in fact no way to define a global energy–momentum vector in a general curved spacetime.

Einstein field equations

In general relativity, the stress-energy tensor is studied in the context of the Einstein field equations which are often written as

where is the Ricci tensor, is the Ricci scalar (the tensor contraction of the Ricci tensor), is the metric tensor, Λ is the cosmological constant (negligible at the scale of a galaxy or smaller), and is the universal gravitational constant.

Stress–energy in special situations

Isolated particle

In special relativity, the stress–energy of a non-interacting particle with rest mass m and trajectory is:

where is the velocity vector (which should not be confused with four-velocity, since it is missing a )

is the Dirac delta function and is the energy of the particle.

Written in language of classical physics, the stress-energy tensor would be (relativistic mass, momentum, the dyad product of momentum and velocity)

.

Stress–energy of a fluid in equilibrium

For a perfect fluid in thermodynamic equilibrium, the stress–energy tensor takes on a particularly simple form

where is the mass–energy density (kilograms per cubic meter), is the hydrostatic pressure (pascals), is the fluid's four velocity, and is the reciprocal of the metric tensor. Therefore, the trace is given by

The four-velocity satisfies

In an inertial frame of reference comoving with the fluid, better known as the fluid's proper frame of reference, the four velocity is

the reciprocal of the metric tensor is simply

and the stress–energy tensor is a diagonal matrix

Electromagnetic stress–energy tensor

The Hilbert stress–energy tensor of a source-free electromagnetic field is

where is the electromagnetic field tensor.

Scalar field

The stress–energy tensor for a complex scalar field which satisfies the Klein–Gordon equation is

and when the metric is flat (Minkowski in Cartesian coordinates) its components work out to be:

Variant definitions of stress–energy

There are a number of inequivalent definitions of non-gravitational stress–energy:

Hilbert stress–energy tensor

The Hilbert stress–energy tensor is defined as the functional derivative

where is the nongravitational part of the action, is the nongravitational part of the Lagrangian density, and the Euler-Lagrange equation has been used. This is symmetric and gauge-invariant. See Einstein–Hilbert action for more information.

Canonical stress–energy tensor

Noether's theorem implies that there is a conserved current associated with translations through space and time. This is called the canonical stress–energy tensor. Generally, this is not symmetric and if we have some gauge theory, it may not be gauge invariant because space-dependent gauge transformations do not commute with spatial translations.

In general relativity, the translations are with respect to the coordinate system and as such, do not transform covariantly. See the section below on the gravitational stress–energy pseudo-tensor.

Belinfante–Rosenfeld stress–energy tensor

In the presence of spin or other intrinsic angular momentum, the canonical Noether stress energy tensor fails to be symmetric. The Belinfante–Rosenfeld stress energy tensor is constructed from the canonical stress–energy tensor and the spin current in such a way as to be symmetric and still conserved. In general relativity, this modified tensor agrees with the Hilbert stress–energy tensor.

Gravitational stress–energy

By the equivalence principle gravitational stress–energy will always vanish locally at any chosen point in some chosen frame, therefore gravitational stress–energy cannot be expressed as a non-zero tensor; instead we have to use a pseudotensor.

In general relativity, there are many possible distinct definitions of the gravitational stress–energy–momentum pseudotensor. These include the Einstein pseudotensor and the Landau–Lifshitz pseudotensor. The Landau–Lifshitz pseudotensor can be reduced to zero at any event in spacetime by choosing an appropriate coordinate system.

Vacuum

From Wikipedia, the free encyclopedia

Pump to demonstrate vacuum

A vacuum is a space devoid of matter. The word is derived from the Latin adjective vacuus for "vacant" or "void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressure. Physicists often discuss ideal test results that would occur in a perfect vacuum, which they sometimes simply call "vacuum" or free space, and use the term partial vacuum to refer to an actual imperfect vacuum as one might have in a laboratory or in space. In engineering and applied physics on the other hand, vacuum refers to any space in which the pressure is considerably lower than atmospheric pressure. The Latin term in vacuo is used to describe an object that is surrounded by a vacuum.

The quality of a partial vacuum refers to how closely it approaches a perfect vacuum. Other things equal, lower gas pressure means higher-quality vacuum. For example, a typical vacuum cleaner produces enough suction to reduce air pressure by around 20%. But higher-quality vacuums are possible. Ultra-high vacuum chambers, common in chemistry, physics, and engineering, operate below one trillionth (10−12) of atmospheric pressure (100 nPa), and can reach around 100 particles/cm3. Outer space is an even higher-quality vacuum, with the equivalent of just a few hydrogen atoms per cubic meter on average in intergalactic space.

Vacuum has been a frequent topic of philosophical debate since ancient Greek times, but was not studied empirically until the 17th century. Evangelista Torricelli produced the first laboratory vacuum in 1643, and other experimental techniques were developed as a result of his theories of atmospheric pressure. A Torricellian vacuum is created by filling a tall glass container closed at one end with mercury, and then inverting it in a bowl to contain the mercury (see below).

Vacuum became a valuable industrial tool in the 20th century with the introduction of incandescent light bulbs and vacuum tubes, and a wide array of vacuum technologies has since become available. The development of human spaceflight has raised interest in the impact of vacuum on human health, and on life forms in general.

Etymology

The word vacuum comes from Latin 'an empty space, void', noun use of neuter of vacuus, meaning "empty", related to vacare, meaning "to be empty".

Vacuum is one of the few words in the English language that contains two consecutive letters u.

Historical understanding

Historically, there has been much dispute over whether such a thing as a vacuum can exist. Ancient Greek philosophers debated the existence of a vacuum, or void, in the context of atomism, which posited void and atom as the fundamental explanatory elements of physics. Following Plato, even the abstract concept of a featureless void faced considerable skepticism: it could not be apprehended by the senses, it could not, itself, provide additional explanatory power beyond the physical volume with which it was commensurate and, by definition, it was quite literally nothing at all, which cannot rightly be said to exist. Aristotle believed that no void could occur naturally, because the denser surrounding material continuum would immediately fill any incipient rarity that might give rise to a void.

In his Physics, book IV, Aristotle offered numerous arguments against the void: for example, that motion through a medium which offered no impediment could continue ad infinitum, there being no reason that something would come to rest anywhere in particular. Although Lucretius argued for the existence of vacuum in the first century BC and Hero of Alexandria tried unsuccessfully to create an artificial vacuum in the first century AD.

In the medieval Muslim world, the physicist and Islamic scholar, Al-Farabi (Alpharabius, 872–950), conducted a small experiment concerning the existence of vacuum, in which he investigated handheld plungers in water. He concluded that air's volume can expand to fill available space, and he suggested that the concept of perfect vacuum was incoherent. According to Nader El-Bizri, the physicist Ibn al-Haytham (Alhazen, 965–1039) and the Mu'tazili theologians disagreed with Aristotle and Al-Farabi, and they supported the existence of a void. Using geometry, Ibn al-Haytham mathematically demonstrated that place (al-makan) is the imagined three-dimensional void between the inner surfaces of a containing body. According to Ahmad Dallal, Abū Rayhān al-Bīrūnī also states that "there is no observable evidence that rules out the possibility of vacuum". The suction pump was described by Arab engineer Al-Jazari in the 13th century, and later appeared in Europe from the 15th century.

European scholars such as Roger Bacon, Blasius of Parma and Walter Burley in the 13th and 14th century focused considerable attention on issues concerning the concept of a vacuum. Eventually following Stoic physics in this instance, scholars from the 14th century onward increasingly departed from the Aristotelian perspective in favor of a supernatural void beyond the confines of the cosmos itself, a conclusion widely acknowledged by the 17th century, which helped to segregate natural and theological concerns.

Almost two thousand years after Plato, René Descartes also proposed a geometrically based alternative theory of atomism, without the problematic nothing–everything dichotomy of void and atom. Although Descartes agreed with the contemporary position, that a vacuum does not occur in nature, the success of his namesake coordinate system and more implicitly, the spatial–corporeal component of his metaphysics would come to define the philosophically modern notion of empty space as a quantified extension of volume. By the ancient definition however, directional information and magnitude were conceptually distinct.

Torricelli's mercury barometer produced one of the first sustained vacuums in a laboratory.

Medieval thought experiments into the idea of a vacuum considered whether a vacuum was present, if only for an instant, between two flat plates when they were rapidly separated. There was much discussion of whether the air moved in quickly enough as the plates were separated, or, as Walter Burley postulated, whether a 'celestial agent' prevented the vacuum arising. The commonly held view that nature abhorred a vacuum was called horror vacui. There was even speculation that even God could not create a vacuum if he wanted and the 1277 Paris condemnations of Bishop Etienne Tempier, which required there to be no restrictions on the powers of God, led to the conclusion that God could create a vacuum if he so wished. Jean Buridan reported in the 14th century that teams of ten horses could not pull open bellows when the port was sealed.

The Crookes tube, used to discover and study cathode rays, was an evolution of the Geissler tube.

The 17th century saw the first attempts to quantify measurements of partial vacuum. Evangelista Torricelli's mercury barometer of 1643 and Blaise Pascal's experiments both demonstrated a partial vacuum.

In 1654, Otto von Guericke invented the first vacuum pump and conducted his famous Magdeburg hemispheres experiment, showing that, owing to atmospheric pressure outside the hemispheres, teams of horses could not separate two hemispheres from which the air had been partially evacuated. Robert Boyle improved Guericke's design and with the help of Robert Hooke further developed vacuum pump technology. Thereafter, research into the partial vacuum lapsed until 1850 when August Toepler invented the Toepler Pump and in 1855 when Heinrich Geissler invented the mercury displacement pump, achieving a partial vacuum of about 10 Pa (0.1 Torr). A number of electrical properties become observable at this vacuum level, which renewed interest in further research.

While outer space provides the most rarefied example of a naturally occurring partial vacuum, the heavens were originally thought to be seamlessly filled by a rigid indestructible material called aether. Borrowing somewhat from the pneuma of Stoic physics, aether came to be regarded as the rarefied air from which it took its name. Early theories of light posited a ubiquitous terrestrial and celestial medium through which light propagated. Additionally, the concept informed Isaac Newton's explanations of both refraction and of radiant heat. 19th century experiments into this luminiferous aether attempted to detect a minute drag on the Earth's orbit. While the Earth does, in fact, move through a relatively dense medium in comparison to that of interstellar space, the drag is so minuscule that it could not be detected. In 1912, astronomer Henry Pickering commented: "While the interstellar absorbing medium may be simply the ether, [it] is characteristic of a gas, and free gaseous molecules are certainly there".

Later, in 1930, Paul Dirac proposed a model of the vacuum as an infinite sea of particles possessing negative energy, called the Dirac sea. This theory helped refine the predictions of his earlier formulated Dirac equation, and successfully predicted the existence of the positron, confirmed two years later. Werner Heisenberg's uncertainty principle, formulated in 1927, predicted a fundamental limit within which instantaneous position and momentum, or energy and time can be measured. This has far reaching consequences on the "emptiness" of space between particles. In the late 20th century, so-called virtual particles that arise spontaneously from empty space were confirmed.

Classical field theories

The strictest criterion to define a vacuum is a region of space and time where all the components of the stress–energy tensor are zero. This means that this region is devoid of energy and momentum, and by consequence, it must be empty of particles and other physical fields (such as electromagnetism) that contain energy and momentum.

Gravity

In general relativity, a vanishing stress–energy tensor implies, through Einstein field equations, the vanishing of all the components of the Ricci tensor. Vacuum does not mean that the curvature of space-time is necessarily flat: the gravitational field can still produce curvature in a vacuum in the form of tidal forces and gravitational waves (technically, these phenomena are the components of the Weyl tensor). The black hole (with zero electric charge) is an elegant example of a region completely "filled" with vacuum, but still showing a strong curvature.

Electromagnetism

In classical electromagnetism, the vacuum of free space, or sometimes just free space or perfect vacuum, is a standard reference medium for electromagnetic effects. Some authors refer to this reference medium as classical vacuum, a terminology intended to separate this concept from QED vacuum or QCD vacuum, where vacuum fluctuations can produce transient virtual particle densities and a relative permittivity and relative permeability that are not identically unity.

In the theory of classical electromagnetism, free space has the following properties:

The vacuum of classical electromagnetism can be viewed as an idealized electromagnetic medium with the constitutive relations in SI units:

relating the electric displacement field D to the electric field E and the magnetic field or H-field H to the magnetic induction or B-field B. Here r is a spatial location and t is time.

Quantum mechanics

A video of an experiment showing vacuum fluctuations (in the red ring) amplified by spontaneous parametric down-conversion.

In quantum mechanics and quantum field theory, the vacuum is defined as the state (that is, the solution to the equations of the theory) with the lowest possible energy (the ground state of the Hilbert space). In quantum electrodynamics this vacuum is referred to as 'QED vacuum' to distinguish it from the vacuum of quantum chromodynamics, denoted as QCD vacuum. QED vacuum is a state with no matter particles (hence the name), and no photons. As described above, this state is impossible to achieve experimentally. (Even if every matter particle could somehow be removed from a volume, it would be impossible to eliminate all the blackbody photons.) Nonetheless, it provides a good model for realizable vacuum, and agrees with a number of experimental observations as described next.

QED vacuum has interesting and complex properties. In QED vacuum, the electric and magnetic fields have zero average values, but their variances are not zero. As a result, QED vacuum contains vacuum fluctuations (virtual particles that hop into and out of existence), and a finite energy called vacuum energy. Vacuum fluctuations are an essential and ubiquitous part of quantum field theory. Some experimentally verified effects of vacuum fluctuations include spontaneous emission and the Lamb shift. Coulomb's law and the electric potential in vacuum near an electric charge are modified.

Theoretically, in QCD multiple vacuum states can coexist. The starting and ending of cosmological inflation is thought to have arisen from transitions between different vacuum states. For theories obtained by quantization of a classical theory, each stationary point of the energy in the configuration space gives rise to a single vacuum. String theory is believed to have a huge number of vacua – the so-called string theory landscape.

Outer space

Structure of the magnetosphere - is not a perfect vacuum, but a tenuous plasma awash with charged particles, free elements such as hydrogen, helium and oxygen, electromagnetic fields.

Outer space has very low density and pressure, and is the closest physical approximation of a perfect vacuum. But no vacuum is truly perfect, not even in interstellar space, where there are still a few hydrogen atoms per cubic meter.

Stars, planets, and moons keep their atmospheres by gravitational attraction, and as such, atmospheres have no clearly delineated boundary: the density of atmospheric gas simply decreases with distance from the object. The Earth's atmospheric pressure drops to about 32 millipascals (4.6×10−6 psi) at 100 kilometres (62 mi) of altitude, the Kármán line, which is a common definition of the boundary with outer space. Beyond this line, isotropic gas pressure rapidly becomes insignificant when compared to radiation pressure from the Sun and the dynamic pressure of the solar winds, so the definition of pressure becomes difficult to interpret. The thermosphere in this range has large gradients of pressure, temperature and composition, and varies greatly due to space weather. Astrophysicists prefer to use number density to describe these environments, in units of particles per cubic centimetre.

But although it meets the definition of outer space, the atmospheric density within the first few hundred kilometers above the Kármán line is still sufficient to produce significant drag on satellites. Most artificial satellites operate in this region called low Earth orbit and must fire their engines every couple of weeks or a few times a year (depending on solar activity). The drag here is low enough that it could theoretically be overcome by radiation pressure on solar sails, a proposed propulsion system for interplanetary travel. Planets are too massive for their trajectories to be significantly affected by these forces, although their atmospheres are eroded by the solar winds.

All of the observable universe is filled with large numbers of photons, the so-called cosmic background radiation, and quite likely a correspondingly large number of neutrinos. The current temperature of this radiation is about 3 K (−270.15 °C; −454.27 °F).

Measurement

The quality of a vacuum is indicated by the amount of matter remaining in the system, so that a high quality vacuum is one with very little matter left in it. Vacuum is primarily measured by its absolute pressure, but a complete characterization requires further parameters, such as temperature and chemical composition. One of the most important parameters is the mean free path (MFP) of residual gases, which indicates the average distance that molecules will travel between collisions with each other. As the gas density decreases, the MFP increases, and when the MFP is longer than the chamber, pump, spacecraft, or other objects present, the continuum assumptions of fluid mechanics do not apply. This vacuum state is called high vacuum, and the study of fluid flows in this regime is called particle gas dynamics. The MFP of air at atmospheric pressure is very short, 70 nm, but at 100 mPa (~1×10−3 Torr) the MFP of room temperature air is roughly 100 mm, which is on the order of everyday objects such as vacuum tubes. The Crookes radiometer turns when the MFP is larger than the size of the vanes.

Vacuum quality is subdivided into ranges according to the technology required to achieve it or measure it. These ranges do not have universally agreed definitions, but a typical distribution is shown in the following table. As we travel into orbit, outer space and ultimately intergalactic space, the pressure varies by several orders of magnitude.

Pressure ranges of each quality of vacuum in different units
Vacuum quality Torr Pa Atmosphere
Atmospheric pressure 760 1.013×105 1
Low vacuum 760 to 25 1×105 to 3×103 9.87×10−1 to 3×10−2
Medium vacuum 25 to 1×10−3 3×103 to 1×10−1 3×10−2 to 9.87×10−7
High vacuum 1×10−3 to 1×10−9 1×10−1 to 1×10−7 9.87×10−7 to 9.87×10−13
Ultra high vacuum 1×10−9 to 1×10−12 1×10−7 to 1×10−10 9.87×10−13 to 9.87×10−16
Extremely high vacuum < 1×10−12 < 1×10−10 < 9.87×10−16
Outer space 1×10−6 to < 1×10−17 1×10−4 to < 3×10−15 9.87×10−10 to < 2.96×10−20
Perfect vacuum 0 0 0
  • Atmospheric pressure is variable but standardized at 101.325 kPa (760 Torr).
  • Low vacuum, also called rough vacuum or coarse vacuum, is vacuum that can be achieved or measured with rudimentary equipment such as a vacuum cleaner and a liquid column manometer.
  • Medium vacuum is vacuum that can be achieved with a single pump, but the pressure is too low to measure with a liquid or mechanical manometer. It can be measured with a McLeod gauge, thermal gauge or a capacitive gauge.
  • High vacuum is vacuum where the MFP of residual gases is longer than the size of the chamber or of the object under test. High vacuum usually requires multi-stage pumping and ion gauge measurement. Some texts differentiate between high vacuum and very high vacuum.
  • Ultra high vacuum requires baking the chamber to remove trace gases, and other special procedures. British and German standards define ultra high vacuum as pressures below 10−6 Pa (10−8 Torr).
  • Deep space is generally much more empty than any artificial vacuum. It may or may not meet the definition of high vacuum above, depending on what region of space and astronomical bodies are being considered. For example, the MFP of interplanetary space is smaller than the size of the Solar System, but larger than small planets and moons. As a result, solar winds exhibit continuum flow on the scale of the Solar System, but must be considered a bombardment of particles with respect to the Earth and Moon.
  • Perfect vacuum is an ideal state of no particles at all. It cannot be achieved in a laboratory, although there may be small volumes which, for a brief moment, happen to have no particles of matter in them. Even if all particles of matter were removed, there would still be photons and gravitons, as well as dark energy, virtual particles, and other aspects of the quantum vacuum.
  • Hard vacuum and soft vacuum are terms that are defined with a dividing line defined differently by different sources, such as 1 Torr, or 0.1 Torr, the common denominator being that a hard vacuum is a higher vacuum than a soft one.

Relative versus absolute measurement

Vacuum is measured in units of pressure, typically as a subtraction relative to ambient atmospheric pressure on Earth. But the amount of relative measurable vacuum varies with local conditions. On the surface of Venus, where ground level atmospheric pressure is much higher than on Earth, much higher relative vacuum readings would be possible. On the surface of the moon with almost no atmosphere, it would be extremely difficult to create a measurable vacuum relative to the local environment.

Similarly, much higher than normal relative vacuum readings are possible deep in the Earth's ocean. A submarine maintaining an internal pressure of 1 atmosphere submerged to a depth of 10 atmospheres (98 metres; a 9.8 metre column of seawater has the equivalent weight of 1 atm) is effectively a vacuum chamber keeping out the crushing exterior water pressures, though the 1 atm inside the submarine would not normally be considered a vacuum.

Therefore, to properly understand the following discussions of vacuum measurement, it is important that the reader assumes the relative measurements are being done on Earth at sea level, at exactly 1 atmosphere of ambient atmospheric pressure.

Measurements relative to 1 atm

A glass McLeod gauge, drained of mercury

The SI unit of pressure is the pascal (symbol Pa), but vacuum is often measured in torrs, named for an Italian physicist Torricelli (1608–1647). A torr is equal to the displacement of a millimeter of mercury (mmHg) in a manometer with 1 torr equaling 133.3223684 pascals above absolute zero pressure. Vacuum is often also measured on the barometric scale or as a percentage of atmospheric pressure in bars or atmospheres. Low vacuum is often measured in millimeters of mercury (mmHg) or pascals (Pa) below standard atmospheric pressure. "Below atmospheric" means that the absolute pressure is equal to the current atmospheric pressure.

In other words, most low vacuum gauges that read, for example 50.79 Torr. Many inexpensive low vacuum gauges have a margin of error and may report a vacuum of 0 Torr but in practice this generally requires a two-stage rotary vane or other medium type of vacuum pump to go much beyond (lower than) 1 torr.

Measuring instruments

Many devices are used to measure the pressure in a vacuum, depending on what range of vacuum is needed.

Hydrostatic gauges (such as the mercury column manometer) consist of a vertical column of liquid in a tube whose ends are exposed to different pressures. The column will rise or fall until its weight is in equilibrium with the pressure differential between the two ends of the tube. The simplest design is a closed-end U-shaped tube, one side of which is connected to the region of interest. Any fluid can be used, but mercury is preferred for its high density and low vapour pressure. Simple hydrostatic gauges can measure pressures ranging from 1 torr (100 Pa) to above atmospheric. An important variation is the McLeod gauge which isolates a known volume of vacuum and compresses it to multiply the height variation of the liquid column. The McLeod gauge can measure vacuums as high as 10−6 torr (0.1 mPa), which is the lowest direct measurement of pressure that is possible with current technology. Other vacuum gauges can measure lower pressures, but only indirectly by measurement of other pressure-controlled properties. These indirect measurements must be calibrated via a direct measurement, most commonly a McLeod gauge.

The kenotometer is a particular type of hydrostatic gauge, typically used in power plants using steam turbines. The kenotometer measures the vacuum in the steam space of the condenser, that is, the exhaust of the last stage of the turbine.

Mechanical or elastic gauges depend on a Bourdon tube, diaphragm, or capsule, usually made of metal, which will change shape in response to the pressure of the region in question. A variation on this idea is the capacitance manometer, in which the diaphragm makes up a part of a capacitor. A change in pressure leads to the flexure of the diaphragm, which results in a change in capacitance. These gauges are effective from 103 torr to 10−4 torr, and beyond.

Thermal conductivity gauges rely on the fact that the ability of a gas to conduct heat decreases with pressure. In this type of gauge, a wire filament is heated by running current through it. A thermocouple or Resistance Temperature Detector (RTD) can then be used to measure the temperature of the filament. This temperature is dependent on the rate at which the filament loses heat to the surrounding gas, and therefore on the thermal conductivity. A common variant is the Pirani gauge which uses a single platinum filament as both the heated element and RTD. These gauges are accurate from 10 torr to 10−3 torr, but they are sensitive to the chemical composition of the gases being measured.

Ionization gauges are used in ultrahigh vacuum. They come in two types: hot cathode and cold cathode. In the hot cathode version an electrically heated filament produces an electron beam. The electrons travel through the gauge and ionize gas molecules around them. The resulting ions are collected at a negative electrode. The current depends on the number of ions, which depends on the pressure in the gauge. Hot cathode gauges are accurate from 10−3 torr to 10−10 torr. The principle behind cold cathode version is the same, except that electrons are produced in a discharge created by a high voltage electrical discharge. Cold cathode gauges are accurate from 10−2 torr to 10−9 torr. Ionization gauge calibration is very sensitive to construction geometry, chemical composition of gases being measured, corrosion and surface deposits. Their calibration can be invalidated by activation at atmospheric pressure or low vacuum. The composition of gases at high vacuums will usually be unpredictable, so a mass spectrometer must be used in conjunction with the ionization gauge for accurate measurement.

Uses

Light bulbs contain a partial vacuum, usually backfilled with argon, which protects the tungsten filament

Vacuum is useful in a variety of processes and devices. Its first widespread use was in the incandescent light bulb to protect the filament from chemical degradation. The chemical inertness produced by a vacuum is also useful for electron beam welding, cold welding, vacuum packing and vacuum frying. Ultra-high vacuum is used in the study of atomically clean substrates, as only a very good vacuum preserves atomic-scale clean surfaces for a reasonably long time (on the order of minutes to days). High to ultra-high vacuum removes the obstruction of air, allowing particle beams to deposit or remove materials without contamination. This is the principle behind chemical vapor deposition, physical vapor deposition, and dry etching which are essential to the fabrication of semiconductors and optical coatings, and to surface science. The reduction of convection provides the thermal insulation of thermos bottles. Deep vacuum lowers the boiling point of liquids and promotes low temperature outgassing which is used in freeze drying, adhesive preparation, distillation, metallurgy, and process purging. The electrical properties of vacuum make electron microscopes and vacuum tubes possible, including cathode ray tubes. Vacuum interrupters are used in electrical switchgear. Vacuum arc processes are industrially important for production of certain grades of steel or high purity materials. The elimination of air friction is useful for flywheel energy storage and ultracentrifuges.

This shallow water well pump reduces atmospheric air pressure inside the pump chamber. Atmospheric pressure extends down into the well, and forces water up the pipe into the pump to balance the reduced pressure. Above-ground pump chambers are only effective to a depth of approximately 9 meters due to the water column weight balancing the atmospheric pressure.

Vacuum-driven machines

Vacuums are commonly used to produce suction, which has an even wider variety of applications. The Newcomen steam engine used vacuum instead of pressure to drive a piston. In the 19th century, vacuum was used for traction on Isambard Kingdom Brunel's experimental atmospheric railway. Vacuum brakes were once widely used on trains in the UK but, except on heritage railways, they have been replaced by air brakes.

Manifold vacuum can be used to drive accessories on automobiles. The best known application is the vacuum servo, used to provide power assistance for the brakes. Obsolete applications include vacuum-driven windscreen wipers and Autovac fuel pumps. Some aircraft instruments (Attitude Indicator (AI) and the Heading Indicator (HI)) are typically vacuum-powered, as protection against loss of all (electrically powered) instruments, since early aircraft often did not have electrical systems, and since there are two readily available sources of vacuum on a moving aircraft, the engine and an external venturi. Vacuum induction melting uses electromagnetic induction within a vacuum.

Maintaining a vacuum in the condenser is an important aspect of the efficient operation of steam turbines. A steam jet ejector or liquid ring vacuum pump is used for this purpose. The typical vacuum maintained in the condenser steam space at the exhaust of the turbine (also called condenser backpressure) is in the range 5 to 15 kPa (absolute), depending on the type of condenser and the ambient conditions.

Outgassing

Evaporation and sublimation into a vacuum is called outgassing. All materials, solid or liquid, have a small vapour pressure, and their outgassing becomes important when the vacuum pressure falls below this vapour pressure. Outgassing has the same effect as a leak and will limit the achievable vacuum. Outgassing products may condense on nearby colder surfaces, which can be troublesome if they obscure optical instruments or react with other materials. This is of great concern to space missions, where an obscured telescope or solar cell can ruin an expensive mission.

The most prevalent outgassing product in vacuum systems is water absorbed by chamber materials. It can be reduced by desiccating or baking the chamber, and removing absorbent materials. Outgassed water can condense in the oil of rotary vane pumps and reduce their net speed drastically if gas ballasting is not used. High vacuum systems must be clean and free of organic matter to minimize outgassing.

Ultra-high vacuum systems are usually baked, preferably under vacuum, to temporarily raise the vapour pressure of all outgassing materials and boil them off. Once the bulk of the outgassing materials are boiled off and evacuated, the system may be cooled to lower vapour pressures and minimize residual outgassing during actual operation. Some systems are cooled well below room temperature by liquid nitrogen to shut down residual outgassing and simultaneously cryopump the system.

Pumping and ambient air pressure

Deep wells have the pump chamber down in the well close to the water surface, or in the water. A "sucker rod" extends from the handle down the center of the pipe deep into the well to operate the plunger. The pump handle acts as a heavy counterweight against both the sucker rod weight and the weight of the water column standing on the upper plunger up to ground level.
 

Fluids cannot generally be pulled, so a vacuum cannot be created by suction. Suction can spread and dilute a vacuum by letting a higher pressure push fluids into it, but the vacuum has to be created first before suction can occur. The easiest way to create an artificial vacuum is to expand the volume of a container. For example, the diaphragm muscle expands the chest cavity, which causes the volume of the lungs to increase. This expansion reduces the pressure and creates a partial vacuum, which is soon filled by air pushed in by atmospheric pressure.

To continue evacuating a chamber indefinitely without requiring infinite growth, a compartment of the vacuum can be repeatedly closed off, exhausted, and expanded again. This is the principle behind positive displacement pumps, like the manual water pump for example. Inside the pump, a mechanism expands a small sealed cavity to create a vacuum. Because of the pressure differential, some fluid from the chamber (or the well, in our example) is pushed into the pump's small cavity. The pump's cavity is then sealed from the chamber, opened to the atmosphere, and squeezed back to a minute size.

A cutaway view of a turbomolecular pump, a momentum transfer pump used to achieve high vacuum

The above explanation is merely a simple introduction to vacuum pumping, and is not representative of the entire range of pumps in use. Many variations of the positive displacement pump have been developed, and many other pump designs rely on fundamentally different principles. Momentum transfer pumps, which bear some similarities to dynamic pumps used at higher pressures, can achieve much higher quality vacuums than positive displacement pumps. Entrapment pumps can capture gases in a solid or absorbed state, often with no moving parts, no seals and no vibration. None of these pumps are universal; each type has important performance limitations. They all share a difficulty in pumping low molecular weight gases, especially hydrogen, helium, and neon.

The lowest pressure that can be attained in a system is also dependent on many things other than the nature of the pumps. Multiple pumps may be connected in series, called stages, to achieve higher vacuums. The choice of seals, chamber geometry, materials, and pump-down procedures will all have an impact. Collectively, these are called vacuum technique. And sometimes, the final pressure is not the only relevant characteristic. Pumping systems differ in oil contamination, vibration, preferential pumping of certain gases, pump-down speeds, intermittent duty cycle, reliability, or tolerance to high leakage rates.

In ultra high vacuum systems, some very "odd" leakage paths and outgassing sources must be considered. The water absorption of aluminium and palladium becomes an unacceptable source of outgassing, and even the adsorptivity of hard metals such as stainless steel or titanium must be considered. Some oils and greases will boil off in extreme vacuums. The permeability of the metallic chamber walls may have to be considered, and the grain direction of the metallic flanges should be parallel to the flange face.

The lowest pressures currently achievable in laboratory are about 1×10−13 torrs (13 pPa). However, pressures as low as 5×10−17 torrs (6.7 fPa) have been indirectly measured in a 4 K (−269.15 °C; −452.47 °F) cryogenic vacuum system. This corresponds to ≈100 particles/cm3.

Effects on humans and animals

This painting, An Experiment on a Bird in the Air Pump by Joseph Wright of Derby, 1768, depicts an experiment performed by Robert Boyle in 1660.

Humans and animals exposed to vacuum will lose consciousness after a few seconds and die of hypoxia within minutes, but the symptoms are not nearly as graphic as commonly depicted in media and popular culture. The reduction in pressure lowers the temperature at which blood and other body fluids boil, but the elastic pressure of blood vessels ensures that this boiling point remains above the internal body temperature of 37 °C. Although the blood will not boil, the formation of gas bubbles in bodily fluids at reduced pressures, known as ebullism, is still a concern. The gas may bloat the body to twice its normal size and slow circulation, but tissues are elastic and porous enough to prevent rupture. Swelling and ebullism can be restrained by containment in a flight suit. Shuttle astronauts wore a fitted elastic garment called the Crew Altitude Protection Suit (CAPS) which prevents ebullism at pressures as low as 2 kPa (15 Torr). Rapid boiling will cool the skin and create frost, particularly in the mouth, but this is not a significant hazard.

Animal experiments show that rapid and complete recovery is normal for exposures shorter than 90 seconds, while longer full-body exposures are fatal and resuscitation has never been successful. A study by NASA on eight chimpanzees found all of them survived two and a half minute exposures to vacuum. There is only a limited amount of data available from human accidents, but it is consistent with animal data. Limbs may be exposed for much longer if breathing is not impaired. Robert Boyle was the first to show in 1660 that vacuum is lethal to small animals.

An experiment indicates that plants are able to survive in a low pressure environment (1.5 kPa) for about 30 minutes.

Cold or oxygen-rich atmospheres can sustain life at pressures much lower than atmospheric, as long as the density of oxygen is similar to that of standard sea-level atmosphere. The colder air temperatures found at altitudes of up to 3 km generally compensate for the lower pressures there. Above this altitude, oxygen enrichment is necessary to prevent altitude sickness in humans that did not undergo prior acclimatization, and spacesuits are necessary to prevent ebullism above 19 km. Most spacesuits use only 20 kPa (150 Torr) of pure oxygen. This pressure is high enough to prevent ebullism, but decompression sickness and gas embolisms can still occur if decompression rates are not managed.

Rapid decompression can be much more dangerous than vacuum exposure itself. Even if the victim does not hold his or her breath, venting through the windpipe may be too slow to prevent the fatal rupture of the delicate alveoli of the lungs. Eardrums and sinuses may be ruptured by rapid decompression, soft tissues may bruise and seep blood, and the stress of shock will accelerate oxygen consumption leading to hypoxia. Injuries caused by rapid decompression are called barotrauma. A pressure drop of 13 kPa (100 Torr), which produces no symptoms if it is gradual, may be fatal if it occurs suddenly.

Some extremophile microorganisms, such as tardigrades, can survive vacuum conditions for periods of days or weeks.

 

Lie point symmetry

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