Green nanotechnology

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Green_nanotechnology 

 

Green nanotechnology refers to the use of nanotechnology to enhance the environmental sustainability of processes producing negative externalities. It also refers to the use of the products of nanotechnology to enhance sustainability. It includes making green nano-products and using nano-products in support of sustainability.

The word GREEN in the name Green Nanotechnology has dual meaning. On one hand it describes the environment friendly technologies utilized to synthesize particles in nano scale; on the other hand it refers to the nanoparticles synthesis mediated by extracts of chlorophyllus plants. Green nanotechnology has been described as the development of clean technologies, "to minimize potential environmental and human health risks associated with the manufacture and use of nanotechnology products. It also encourages replacement of existing products with new nano-products that are more environmentally friendly throughout their lifecycle."

Aim

Green nanotechnology has two goals: producing nanomaterials and products without harming the environment or human health, and producing nano-products that provide solutions to environmental problems. It uses existing principles of green chemistry and green engineering to make nanomaterials and nano-products without toxic ingredients, at low temperatures using less energy and renewable inputs wherever possible, and using lifecycle thinking in all design and engineering stages.

In addition to making nanomaterials and products with less impact to the environment, green nanotechnology also means using nanotechnology to make current manufacturing processes for non-nano materials and products more environmentally friendly. For example, nanoscale membranes can help separate desired chemical reaction products from waste materials from plants. Nanoscale catalysts can make chemical reactions more efficient and less wasteful. Sensors at the nanoscale can form a part of process control systems, working with nano-enabled information systems. Using alternative energy systems, made possible by nanotechnology, is another way to "green" manufacturing processes.

The second goal of green nanotechnology involves developing products that benefit the environment either directly or indirectly. Nanomaterials or products directly can clean hazardous waste sites, desalinate water, treat pollutants, or sense and monitor environmental pollutants. Indirectly, lightweight nanocomposites for automobiles and other means of transportation could save fuel and reduce materials used for production; nanotechnology-enabled fuel cells and light-emitting diodes (LEDs) could reduce pollution from energy generation and help conserve fossil fuels; self-cleaning nanoscale surface coatings could reduce or eliminate many cleaning chemicals used in regular maintenance routines; and enhanced battery life could lead to less material use and less waste. Green Nanotechnology takes a broad systems view of nanomaterials and products, ensuring that unforeseen consequences are minimized and that impacts are anticipated throughout the full life cycle.

Current research

Solar cells

Main article: Energy applications of nanotechnology

Research is underway to use nanomaterials for purposes including more efficient solar cells, practical fuel cells, and environmentally friendly batteries. The most advanced nanotechnology projects related to energy are: storage, conversion, manufacturing improvements by reducing materials and process rates, energy saving (by better thermal insulation for example), and enhanced renewable energy sources.

One major project that is being worked on is the development of nanotechnology in solar cells. Solar cells are more efficient as they get tinier and solar energy is a renewable resource. The price per watt of solar energy is lower than one dollar.

Research is ongoing to use nanowires and other nanostructured materials with the hope of to create cheaper and more efficient solar cells than are possible with conventional planar silicon solar cells. Another example is the use of fuel cells powered by hydrogen, potentially using a catalyst consisting of carbon supported noble metal particles with diameters of 1–5 nm. Materials with small nanosized pores may be suitable for hydrogen storage. Nanotechnology may also find applications in batteries, where the use of nanomaterials may enable batteries with higher energy content or supercapacitors with a higher rate of recharging.

Nanotechnology is already used to provide improved performance coatings for photovoltaic (PV) and solar thermal panels. Hydrophobic and self-cleaning properties combine to create more efficient solar panels, especially during inclement weather. PV covered with nanotechnology coatings are said to stay cleaner for longer to ensure maximum energy efficiency is maintained.

Nanoremediation and water treatment

Main articles: Nanofiltration and Nanoremediation

Nanotechnology offers the potential of novel nanomaterials for the treatment of surface water, groundwater, wastewater, and other environmental materials contaminated by toxic metal ions, organic and inorganic solutes, and microorganisms. Due to their unique activity toward recalcitrant contaminants, many nanomaterials are under active research and development for use in the treatment of water and contaminated sites.

The present market of nanotech-based technologies applied in water treatment consists of reverse osmosis(RO), nanofiltration, ultrafiltration membranes. Indeed, among emerging products one can name nanofiber filters, carbon nanotubes and various nanoparticles.

Nanotechnology is expected to deal more efficiently with contaminants which convectional water treatment systems struggle to treat, including bacteria, viruses and heavy metals. This efficiency generally stems from the very high specific surface area of nanomaterials, which increases dissolution, reactivity and sorption of contaminants.

Environmental remediation

Main article: Nanoremediation

Nanoremediation is the use of nanoparticles for environmental remediation. Nanoremediation has been most widely used for groundwater treatment, with additional extensive research in wastewater treatment. Nanoremediation has also been tested for soil and sediment cleanup. Even more preliminary research is exploring the use of nanoparticles to remove toxic materials from gases.

Some nanoremediation methods, particularly the use of nano zerovalent iron for groundwater cleanup, have been deployed at full-scale cleanup sites. Nanoremediation is an emerging industry; by 2009, nanoremediation technologies had been documented in at least 44 cleanup sites around the world, predominantly in the United States. During nanoremediation, a nanoparticle agent must be brought into contact with the target contaminant under conditions that allow a detoxifying or immobilizing reaction. This process typically involves a pump-and-treat process or in situ application. Other methods remain in research phases.

Scientists have been researching the capabilities of buckminsterfullerene in controlling pollution, as it may be able to control certain chemical reactions. Buckminsterfullerene has been demonstrated as having the ability of inducing the protection of reactive oxygen species and causing lipid peroxidation. This material may allow for hydrogen fuel to be more accessible to consumers.

Water cleaning technology

In 2017 the RingwooditE Co Ltd was formed in order to explore Thermonuclear Trap Technology (TTT) for the purpose of cleaning all sources of water from pollution and toxic contents. This patented nanotechnology uses a high pressure and temperature chamber to separate isotopes that should by nature not be in drinking water to pure drinking water, as to the by the WHO´s established classification. This method has been developed by among others, by professor Vladimir Afanasiew, at the Moscow Nuclear Institution. This technology is targeted to clean Sea, river, lake and landfill waste waters. It even removes radioactive isotopes from the sea water, after Nuclear Power Stations catastrophes and cooling water plant towers. By this technology pharmaca rests are being removed as well as narcotics and tranquilizers. Bottom layers and sides at lake and rivers can be returned, after being cleaned. Machinery used for this purpose are much similar to those of deep sea mining. Removed waste items are being sorted by the process, and can be re used as raw material for other industrial production.

Water filtration

Main article: Nanofiltration

Nanofiltration is a relatively recent membrane filtration process used most often with low total dissolved solids water such as surface water and fresh groundwater, with the purpose of softening (polyvalent cation removal) and removal of disinfection by-product precursors such as natural organic matter and synthetic organic matter. Nanofiltration is also becoming more widely used in food processing applications such as dairy, for simultaneous concentration and partial (monovalent ion) demineralisation.

Nanofiltration is a membrane filtration based method that uses nanometer sized cylindrical through-pores that pass through the membrane at a 90°. Nanofiltration membranes have pore sizes from 1-10 Angstrom, smaller than that used in microfiltration and ultrafiltration, but just larger than that in reverse osmosis. Membranes used are predominantly created from polymer thin films. Materials that are commonly used include polyethylene terephthalate or metals such as aluminum. Pore dimensions are controlled by pH, temperature and time during development with pore densities ranging from 1 to 106 pores per cm². Membranes made from polyethylene terephthalate and other similar materials, are referred to as "track-etch" membranes, named after the way the pores on the membranes are made. "Tracking" involves bombarding the polymer thin film with high energy particles. This results in making tracks that are chemically developed into the membrane, or "etched" into the membrane, which are the pores. Membranes created from metal such as alumina membranes, are made by electrochemically growing a thin layer of aluminum oxide from aluminum metal in an acidic medium.

Some water-treatment devices incorporating nanotechnology are already on the market, with more in development. Low-cost nanostructured separation membranes methods have been shown to be effective in producing potable water in a recent study.

Nanotech to disinfect water

Nanotechnology provides an alternative solution to clean germs in water, a problem that has been getting worse due to the population explosion, growing need for clean water and the emergence of additional pollutants. One of the alternatives offered is antimicrobial nanotechnology stated that several nanomaterials showed strong antimicrobial properties through diverse mechanisms, such as photocatalytic production of reactive oxygen species that damage cell components and viruses. There is also the case of the synthetically-fabricated nanometallic particles that produce antimicrobial action called oligodynamic disinfection, which can inactivate microorganisms at low concentrations. Commercial purification systems based on titanium oxide photocatalysis also currently exist and studies show that this technology can achieve complete inactivation of fecal coliforms in 15 minutes once activated by sunlight.

There are four classes of nanomaterials that are employed for water treatment and these are dendrimers, zeolites, carbonaceous nanomaterials, and metals containing nanoparticles. The benefits of the reduction of the size of the metals (e.g. silver, copper, titanium, and cobalt) to the nanoscale such as contact efficiency, greater surface area, and better elution properties.

Cleaning up oil spills

The U.S. Environmental Protection Agency (EPA) documents more than ten thousand oil spills per year. Conventionally, biological, dispersing, and gelling agents are deployed to remedy oil spills. Although, these methods have been used for decades, none of these techniques can retrieve the irreplaceable lost oil. However, nanowires can not only swiftly clean up oil spills but also recover as much oil as possible. These nanowires form a mesh that absorbs up to twenty times its weight in hydrophobic liquids while rejecting water with its water repelling coating. Since the potassium manganese oxide is very stable even at high temperatures, the oil can be boiled off the nanowires and both the oil and the nanowires can then be reused.

In 2005, Hurricane Katrina damaged or destroyed more than thirty oil platforms and nine refineries. The Interface Science Corporation successfully launched a new oil remediation and recovery application, which used the water repelling nanowires to clean up the oil spilled by the damaged oil platforms and refineries.

Removing plastics from oceans

One innovation of green nanotechnology that is currently under development are nanomachines modeled after a bacterium bioengineered to consume plastics, Ideonella sakaiensis. These nano-machines are able to decompose plastics dozens of times faster than the bioengineered bacteria not only because of their increased surface area but also because the energy released from decomposing the plastic is used to fuel the nano-machines.

Air pollution control

In addition to water treatment and environmental remediation, nanotechnology is currently improving air quality. Nanoparticles can be engineered to catalyze, or hasten, the reaction to transform environmentally pernicious gases into harmless ones. For example, many industrial factories that produce large amounts harmful gases employ a type of nanofiber catalyst made of magnesium oxide (Mg₂O) to purify dangerous organic substances in the smoke. Although chemical catalysts already exist in the gaseous vapors from cars, nanotechnology has a greater chance of reacting with the harmful substances in the vapors. This greater probability comes from the fact that nanotechnology can interact with more particles because of its greater surface area.

Nanotechnology has been used to remediate air pollution including car exhaust pollution, and potentially greenhouse gases due to its high surface area. Based on research done by the Environmental Science Pollution Research International, nanotechnology can specifically help to treat carbon-based nanoparticles, greenhouse gases, and volatile organic compounds. There is also work being done to develop antibacterial nanoparticles, metal oxide nanoparticles, and amendment agents for phytoremediation processes. Nanotechnology can also give the possibility of preventing air pollution in the first place due to its extremely small scale. Nanotechnology has been accepted as a tool for many industrial and domestic fields like gas monitoring systems, fire and toxic gas detectors, ventilation control, breath alcohol detectors and many more. Other sources state that nanotechnology has the potential to develop the pollutants sensing and detection methods that already exist. The ability to detect pollutants and sense unwanted materials will be heightened by the large surface area of nanomaterials and their high surface energy. The World Health Organization declared in 2014 that air contamination caused around 7 million deaths in 2012. This new technology could be an essential asset to this epidemic. The three ways that nanotechnology is being used to treat air pollution are nano-adsorptive materials, degradation by nanocatalysis, and filtration/separation by nanofilters. Nanoscale adsorbents being the main alleviator for many air pollution difficulties. Their structure permits a great interaction with organic compounds as well as increased selectivity and stability in maximum adsorption capacity. Other advantages include high electrical and thermal conductivities, high strength, high hardness. Target pollutants that can be targeted by nanomolecules are 〖NO〗_x, 〖CO〗_2, 〖NH〗_3, N_2, VOCs, Isopropyl vapor, 〖CH〗_3 OH gases, N_2 O, H_2 S. Carbon nanotubes specifically remove particles in many ways. One method is by passing them through the nanotubes where the molecules are oxidized; the molecules then are adsorbed on a nitrate species. Carbon nanotubes with amine groups provide numerous chemical sites for carbon dioxide adsorption at low temperature ranges of 20°-100° degrees Celsius. Van der Waals forces and π-π interactions also are used to pull molecules onto surface functional groups. Fullerene can be used to rid of carbon dioxide pollution due to its high adsorption capacity. Graphene nanotubes have functional groups that adsorb gases. There are plenty of nanocatalysts that can be used for air pollution reduction and air quality. Some of these materials include 〖TiO〗_2, Vanadium, Platinum, Palladium, Rhodium, and Silver. Catalytic industrial emission reduction, car exhaust reduction, and air purification are just some of the major thrusts that these nanomaterials are being utilized within. Certain applications are not widely spread, but other are more popular. Indoor air pollution is barely on the market yet, but it is being developed more efficiently due to complications with health effects. Car exhaust emission reduction is widely used in diesel fueled automobiles currently being one of the more popular applications. Industrial emission reduction is also widely used. It is n integral method specifically at coal fired power plants as well as refineries. These methods are analyzed and reviewed using SEM imaging to ensure its usefulness and accuracy.

Additionally, research is currently being conducted to find out if nanoparticles can be engineered to separate car exhaust from methane or carbon dioxide, which has been known to damage the Earth's ozone layer. In fact, John Zhu, a professor at the University of Queensland, is exploring the creation of a carbon nanotube(CNT) which can trap greenhouse gases hundreds of times more efficiently than current methods can.

Nanotechnology for sensors

Perpetual exposure to heavy metal pollution and particulate matter will lead to health concerns such as lung cancer, heart conditions, and even motor neuron diseases. However, humanity's ability to shield themselves from these health problems can be improved by accurate and swift nanocontact-sensors able to detect pollutants at the atomic level. These nanocontact sensors do not require much energy to detect metal ions or radioactive elements. Additionally, they can be made in automatic mode so that they can be readably used at any given moment. Additionally, these nanocontact sensors are energy and cost effective since they are composed with conventional microelectronic manufacturing equipment using electrochemical techniques.

Some examples of nano-based monitoring include:

1. Functionalized nanoparticles able to form anionic oxidants bonding thereby allowing the detection of carcinogenic substances at very low concentrations.
2. Polymer nanospheres have been developed to measure organic contaminates in very low concentrations
3. "Peptide nanoelectrodes have been employed based on the concept of thermocouple. In a 'nano-distance separation gap, a peptide molecule is placed to form a molecular junction. When a specific metal ion is bound to the gap; the electrical current will result conductance in a unique value. Hence the metal ion will be easily detected."
4. Composite electrodes, a mixture of nanotubes and copper, have been created to detect substances such as organophosphorus pesticides, carbohydrates and other woods pathogenic substances in low concentrations.

Concerns

Although green nanotechnology poses many advantages over traditional methods, there is still much debate about the concerns brought about by nanotechnology. For example, since the nanoparticles are small enough to be absorbed into skin and/or inhaled, countries are mandating that additional research revolving around the impact of nanotechnology on organisms be heavily studied. In fact, the field of eco-nanotoxicology was founded solely to study the effect of nanotechnology on earth and all of its organisms. At the moment, scientists are unsure of what will happen when nanoparticles seep into soil and water, but organizations, such as NanoImpactNet, have set out to study these effects.

Einstein field equations

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Einstein_field_equations

 

In the 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 within it.

The equations were published by Albert Einstein in 1915 in the form of a tensor equation which related the local spacetime curvature (expressed by the Einstein tensor) with the local energy, momentum and stress within that spacetime (expressed by the stress–energy tensor).

Analogously to the way that electromagnetic fields are related to the distribution of charges and currents via Maxwell's equations, the EFE relate the spacetime geometry to the distribution of mass–energy, momentum and stress, that is, they determine the metric tensor of spacetime for a given arrangement of stress–energy–momentum in the spacetime. The relationship between the metric tensor and the Einstein tensor allows the EFE to be written as a set of nonlinear partial differential equations when used in this way. The solutions of the EFE are the components of the metric tensor. The inertial trajectories of particles and radiation (geodesics) in the resulting geometry are then calculated using the geodesic equation.

As well as implying local energy–momentum conservation, the EFE reduce to Newton's law of gravitation in the limit of a weak gravitational field and velocities that are much less than the speed of light.

Exact solutions for the EFE can only be found under simplifying assumptions such as symmetry. Special classes of exact solutions are most often studied since they model many gravitational phenomena, such as rotating black holes and the expanding universe. Further simplification is achieved in approximating the spacetime as having only small deviations from flat spacetime, leading to the linearized EFE. These equations are used to study phenomena such as gravitational waves.

Mathematical form

 The Einstein field equations (EFE) may be written in the form:

${\displaystyle G_{\mu \nu }+\mathrm{\Lambda }{g}_{\mu \nu }=\kappa T_{\mu \nu }}$

EFE on a wall in Leiden, Netherlands

where ${\displaystyle G_{\mu \nu }}$ is the Einstein tensor, ${\displaystyle g_{\mu \nu }}$ is the metric tensor, ${\displaystyle T_{\mu \nu }}$ is the stress–energy tensor, ${\displaystyle \mathrm{\Lambda }}$ is the cosmological constant and ${\displaystyle \kappa }$ is the Einstein gravitational constant.

The Einstein tensor is defined as

${\displaystyle G_{\mu \nu }=R_{\mu \nu }-\frac{1}{2}R{g}_{\mu \nu },}$

where R_μν is the Ricci curvature tensor, and R is the scalar curvature. This is a symmetric second-degree tensor that depends on only the metric tensor and its first and second derivatives.

The Einstein gravitational constant is defined as

${\displaystyle \kappa =\frac{8\pi G}{c^4}\approx 2.076647442844\times {10}^{-43}{\text{N}}^{-1},}$

where G is the Newtonian constant of gravitation and c is the speed of light in vacuum.

The EFE can thus also be written as

${\displaystyle R_{\mu \nu }-\frac{1}{2}R{g}_{\mu \nu }+\mathrm{\Lambda }{g}_{\mu \nu }=\kappa T_{\mu \nu }.}$

In standard units, each term on the left has units of 1/length².

The expression on the left represents the curvature of spacetime as determined by the metric; the expression on the right represents the stress–energy–momentum content of spacetime. The EFE can then be interpreted as a set of equations dictating how stress–energy–momentum determines the curvature of spacetime.

These equations, together with the geodesic equation, which dictates how freely falling matter moves through spacetime, form the core of the mathematical formulation of general relativity.

The EFE is a tensor equation relating a set of symmetric 4 × 4 tensors. Each tensor has 10 independent components. The four Bianchi identities reduce the number of independent equations from 10 to 6, leaving the metric with four gauge-fixing degrees of freedom, which correspond to the freedom to choose a coordinate system.

Although the Einstein field equations were initially formulated in the context of a four-dimensional theory, some theorists have explored their consequences in n dimensions. The equations in contexts outside of general relativity are still referred to as the Einstein field equations. The vacuum field equations (obtained when T_μν is everywhere zero) define Einstein manifolds.

The equations are more complex than they appear. Given a specified distribution of matter and energy in the form of a stress–energy tensor, the EFE are understood to be equations for the metric tensor ${\displaystyle g_{\mu \nu }}$, since both the Ricci tensor and scalar curvature depend on the metric in a complicated nonlinear manner. When fully written out, the EFE are a system of ten coupled, nonlinear, hyperbolic-elliptic partial differential equations.

Sign convention

The above form of the EFE is the standard established by Misner, Thorne, and Wheeler (MTW). The authors analyzed conventions that exist and classified these according to three signs ([S1] [S2] [S3]):

$${\displaystyle \begin{array}{rl}{g}_{\mu \nu }& =[S1]\times \mathrm{diag}(-1,+1,+1,+1)\\ {R^{\mu }}_{\alpha \beta \gamma }& =[S2]\times \left({\mathrm{\Gamma }}_{\alpha \gamma ,\beta }^{\mu }-{\mathrm{\Gamma }}_{\alpha \beta ,\gamma }^{\mu }+{\mathrm{\Gamma }}_{\sigma \beta }^{\mu }{\mathrm{\Gamma }}_{\gamma \alpha }^{\sigma }-{\mathrm{\Gamma }}_{\sigma \gamma }^{\mu }{\mathrm{\Gamma }}_{\beta \alpha }^{\sigma }\right)\\ G_{\mu \nu }& =[S3]\times \kappa T_{\mu \nu }\end{array}}$$

The third sign above is related to the choice of convention for the Ricci tensor:

$${\displaystyle R_{\mu \nu }=[S2]\times [S3]\times {R^{\alpha }}_{\mu \alpha \nu }}$$

With these definitions Misner, Thorne, and Wheeler classify themselves as (+ + +), whereas Weinberg (1972) is (+ − −), Peebles (1980) and Efstathiou et al. (1990) are (− + +), Rindler (1977), Atwater (1974), Collins Martin & Squires (1989) and Peacock (1999) are (− + −).

Authors including Einstein have used a different sign in their definition for the Ricci tensor which results in the sign of the constant on the right side being negative:

$${\displaystyle R_{\mu \nu }-\frac{1}{2}R{g}_{\mu \nu }-\mathrm{\Lambda }{g}_{\mu \nu }=-\kappa T_{\mu \nu }.}$$

The sign of the cosmological term would change in both these versions if the (+ − − −) metric sign convention is used rather than the MTW (− + + +) metric sign convention adopted here.

Equivalent formulations

Taking the trace with respect to the metric of both sides of the EFE one gets

$${\displaystyle R-\frac{D}{2}R+D\mathrm{\Lambda }=\kappa T,}$$

where D is the spacetime dimension. Solving for R and substituting this in the original EFE, one gets the following equivalent "trace-reversed" form:

$${\displaystyle R_{\mu \nu }-\frac{2}{D-2}\mathrm{\Lambda }{g}_{\mu \nu }=\kappa \left(T_{\mu \nu }-\frac{1}{D-2}T{g}_{\mu \nu }\right).}$$

In D = 4 dimensions this reduces to

$${\displaystyle R_{\mu \nu }-\mathrm{\Lambda }{g}_{\mu \nu }=\kappa \left(T_{\mu \nu }-\frac{1}{2}T{g}_{\mu \nu }\right).}$$

Reversing the trace again would restore the original EFE. The trace-reversed form may be more convenient in some cases (for example, when one is interested in weak-field limit and can replace ${\displaystyle g_{\mu \nu }}$ in the expression on the right with the Minkowski metric without significant loss of accuracy).

The cosmological constant

Main article: Cosmological constant

In the Einstein field equations

$${\displaystyle G_{\mu \nu }+\mathrm{\Lambda }{g}_{\mu \nu }=\kappa T_{\mu \nu },}$$

the term containing the cosmological constant Λ was absent from the version in which he originally published them. Einstein then included the term with the cosmological constant to allow for a universe that is not expanding or contracting. This effort was unsuccessful because:

• any desired steady state solution described by this equation is unstable, and
• observations by Edwin Hubble showed that our universe is expanding.

Einstein then abandoned Λ, remarking to George Gamow "that the introduction of the cosmological term was the biggest blunder of his life".

The inclusion of this term does not create inconsistencies. For many years the cosmological constant was almost universally assumed to be zero. More recent astronomical observations have shown an accelerating expansion of the universe, and to explain this a positive value of Λ is needed. The cosmological constant is negligible at the scale of a galaxy or smaller.

Einstein thought of the cosmological constant as an independent parameter, but its term in the field equation can also be moved algebraically to the other side and incorporated as part of the stress–energy tensor:

$${\displaystyle T_{\mu \nu }^{(\mathrm{v}\mathrm{a}\mathrm{c})}=-\frac{\mathrm{\Lambda }}{\kappa }{g}_{\mu \nu }.}$$

This tensor describes a vacuum state with an energy density ρ_vac and isotropic pressure p_vac that are fixed constants and given by

$${\displaystyle {\rho }_{\mathrm{v}\mathrm{a}\mathrm{c}}=-p_{\mathrm{v}\mathrm{a}\mathrm{c}}=\frac{\mathrm{\Lambda }}{\kappa },}$$

where it is assumed that Λ has SI unit m⁻² and κ is defined as above.

The existence of a cosmological constant is thus equivalent to the existence of a vacuum energy and a pressure of opposite sign. This has led to the terms "cosmological constant" and "vacuum energy" being used interchangeably in general relativity.

Features

Conservation of energy and momentum

General relativity is consistent with the local conservation of energy and momentum expressed as

$${\displaystyle {\mathrm{\nabla }}_{\beta }{T}^{\alpha \beta }={T^{\alpha \beta }}_{;\beta }=0.}$$

Derivation of local energy–momentum conservation

Contracting the differential Bianchi identity

$${\displaystyle R_{\alpha \beta [\gamma \delta ;\epsilon ]}=0}$$

with g^αβ gives, using the fact that the metric tensor is covariantly constant, i.e. g^αβ_;γ = 0,

$${\displaystyle {R^{\gamma }}_{\beta \gamma \delta ;\epsilon }+{R^{\gamma }}_{\beta \epsilon \gamma ;\delta }+{R^{\gamma }}_{\beta \delta \epsilon ;\gamma }=0}$$

The antisymmetry of the Riemann tensor allows the second term in the above expression to be rewritten:

$${\displaystyle {R^{\gamma }}_{\beta \gamma \delta ;\epsilon }-{R^{\gamma }}_{\beta \gamma \epsilon ;\delta }+{R^{\gamma }}_{\beta \delta \epsilon ;\gamma }=0}$$

which is equivalent to

$${\displaystyle R_{\beta \delta ;\epsilon }-R_{\beta \epsilon ;\delta }+{R^{\gamma }}_{\beta \delta \epsilon ;\gamma }=0}$$

using the definition of the Ricci tensor.

Next, contract again with the metric

$${\displaystyle g^{\beta \delta }\left(R_{\beta \delta ;\epsilon }-R_{\beta \epsilon ;\delta }+{R^{\gamma }}_{\beta \delta \epsilon ;\gamma }\right)=0}$$

to get

$${\displaystyle {R^{\delta }}_{\delta ;\epsilon }-{R^{\delta }}_{\epsilon ;\delta }+{R^{\gamma \delta }}_{\delta \epsilon ;\gamma }=0}$$

The definitions of the Ricci curvature tensor and the scalar curvature then show that

$${\displaystyle R_{;\epsilon }-2{R^{\gamma }}_{\epsilon ;\gamma }=0}$$

which can be rewritten as

$${\displaystyle {\left({R^{\gamma }}_{\epsilon }-\frac{1}{2}{g^{\gamma }}_{\epsilon }R\right)}_{;\gamma }=0}$$

A final contraction with g^εδ gives

$${\displaystyle {\left(R^{\gamma \delta }-\frac{1}{2}{g}^{\gamma \delta }R\right)}_{;\gamma }=0}$$

which by the symmetry of the bracketed term and the definition of the Einstein tensor, gives, after relabelling the indices,

$${\displaystyle {G^{\alpha \beta }}_{;\beta }=0}$$

Using the EFE, this immediately gives,

$${\displaystyle {\mathrm{\nabla }}_{\beta }{T}^{\alpha \beta }={T^{\alpha \beta }}_{;\beta }=0}$$

which expresses the local conservation of stress–energy. This conservation law is a physical requirement. With his field equations Einstein ensured that general relativity is consistent with this conservation condition.

Nonlinearity

The nonlinearity of the EFE distinguishes general relativity from many other fundamental physical theories. For example, Maxwell's equations of electromagnetism are linear in the electric and magnetic fields, and charge and current distributions (i.e. the sum of two solutions is also a solution); another example is Schrödinger's equation of quantum mechanics, which is linear in the wavefunction.

The correspondence principle

The EFE reduce to Newton's law of gravity by using both the weak-field approximation and the slow-motion approximation. In fact, the constant G appearing in the EFE is determined by making these two approximations.

Derivation of Newton's law of gravity

Newtonian gravitation can be written as the theory of a scalar field, Φ, which is the gravitational potential in joules per kilogram of the gravitational field g = −∇Φ, see Gauss's law for gravity

$${\displaystyle {\mathrm{\nabla }}^2\mathrm{\Phi }\left(\overrightarrow{x},t\right)=4\pi G\rho \left(\overrightarrow{x},t\right)}$$

where ρ is the mass density. The orbit of a free-falling particle satisfies

$${\displaystyle \ddot{\overrightarrow{x}}(t)=\overrightarrow{g}=-\mathrm{\nabla }\mathrm{\Phi }\left(\overrightarrow{x}(t),t\right).}$$

In tensor notation, these become

$${\displaystyle \begin{array}{rl}{\mathrm{\Phi }}_{,ii}& =4\pi G\rho \\ \frac{d^2{x}^i}{d{t}^2}& =-{\mathrm{\Phi }}_{,i}.\end{array}}$$

In general relativity, these equations are replaced by the Einstein field equations in the trace-reversed form

$${\displaystyle R_{\mu \nu }=K\left(T_{\mu \nu }-\frac{1}{2}T{g}_{\mu \nu }\right)}$$

for some constant, K, and the geodesic equation

$${\displaystyle \frac{d^2{x}^{\alpha }}{d{\tau }^2}=-{\mathrm{\Gamma }}_{\beta \gamma }^{\alpha }\frac{d{x}^{\beta }}{d\tau }\frac{d{x}^{\gamma }}{d\tau }.}$$

To see how the latter reduces to the former, we assume that the test particle's velocity is approximately zero

$${\displaystyle \frac{d{x}^{\beta }}{d\tau }\approx \left(\frac{dt}{d\tau },0,0,0\right)}$$

and thus

$${\displaystyle \frac{d}{dt}\left(\frac{dt}{d\tau }\right)\approx 0}$$

and that the metric and its derivatives are approximately static and that the squares of deviations from the Minkowski metric are negligible. Applying these simplifying assumptions to the spatial components of the geodesic equation gives

$${\displaystyle \frac{d^2{x}^i}{d{t}^2}\approx -{\mathrm{\Gamma }}_{00}^i}$$

where two factors of dt/dτ have been divided out. This will reduce to its Newtonian counterpart, provided

$${\displaystyle {\mathrm{\Phi }}_{,i}\approx {\mathrm{\Gamma }}_{00}^i=\frac{1}{2}{g}^{i\alpha }\left(g_{\alpha 0,0}+g_{0\alpha ,0}-g_{00,\alpha }\right).}$$

Our assumptions force α = i and the time (0) derivatives to be zero. So this simplifies to

$${\displaystyle 2{\mathrm{\Phi }}_{,i}\approx g^{ij}\left(-g_{00,j}\right)\approx -g_{00,i}}$$

which is satisfied by letting

$${\displaystyle g_{00}\approx -c^2-2\mathrm{\Phi }.}$$

Turning to the Einstein equations, we only need the time-time component

$${\displaystyle R_{00}=K\left(T_{00}-\frac{1}{2}T{g}_{00}\right)}$$

the low speed and static field assumptions imply that

$${\displaystyle T_{\mu \nu }\approx \mathrm{diag}\left(T_{00},0,0,0\right)\approx \mathrm{diag}\left(\rho c^4,0,0,0\right).}$$

So

$${\displaystyle T=g^{\alpha \beta }{T}_{\alpha \beta }\approx g^{00}{T}_{00}\approx -\frac{1}{c^2}\rho c^4=-\rho c^2}$$

and thus

$${\displaystyle K\left(T_{00}-\frac{1}{2}T{g}_{00}\right)\approx K\left(\rho c^4-\frac{1}{2}\left(-\rho c^2\right)\left(-c^2\right)\right)=\frac{1}{2}K\rho c^4.}$$

From the definition of the Ricci tensor

$${\displaystyle R_{00}={\mathrm{\Gamma }}_{00,\rho }^{\rho }-{\mathrm{\Gamma }}_{\rho 0,0}^{\rho }+{\mathrm{\Gamma }}_{\rho \lambda }^{\rho }{\mathrm{\Gamma }}_{00}^{\lambda }-{\mathrm{\Gamma }}_{0\lambda }^{\rho }{\mathrm{\Gamma }}_{\rho 0}^{\lambda }.}$$

Our simplifying assumptions make the squares of Γ disappear together with the time derivatives

$${\displaystyle R_{00}\approx {\mathrm{\Gamma }}_{00,i}^i.}$$

Combining the above equations together

$${\displaystyle {\mathrm{\Phi }}_{,ii}\approx {\mathrm{\Gamma }}_{00,i}^i\approx R_{00}=K\left(T_{00}-\frac{1}{2}T{g}_{00}\right)\approx \frac{1}{2}K\rho c^4}$$

which reduces to the Newtonian field equation provided

$${\displaystyle \frac{1}{2}K\rho c^4=4\pi G\rho }$$

which will occur if

$${\displaystyle K=\frac{8\pi G}{c^4}.}$$

Vacuum field equations

A Swiss commemorative coin from 1979, showing the vacuum field equations with zero cosmological constant (top).

If the energy–momentum tensor T_μν is zero in the region under consideration, then the field equations are also referred to as the vacuum field equations. By setting T_μν = 0 in the trace-reversed field equations, the vacuum field equations, also known as 'Einstein vacuum equations' (EVE), can be written as

$${\displaystyle R_{\mu \nu }=0.}$$

In the case of nonzero cosmological constant, the equations are

$${\displaystyle R_{\mu \nu }=\frac{\mathrm{\Lambda }}{\frac{D}{2}-1}{g}_{\mu \nu }.}$$

The solutions to the vacuum field equations are called vacuum solutions. Flat Minkowski space is the simplest example of a vacuum solution. Nontrivial examples include the Schwarzschild solution and the Kerr solution.

Manifolds with a vanishing Ricci tensor, R_μν = 0, are referred to as Ricci-flat manifolds and manifolds with a Ricci tensor proportional to the metric as Einstein manifolds.

Einstein–Maxwell equations

See also: Maxwell's equations in curved spacetime

If the energy–momentum tensor T_μν is that of an electromagnetic field in free space, i.e. if the electromagnetic stress–energy tensor

$${\displaystyle T^{\alpha \beta }=-\frac{1}{{\mu }_0}\left({F^{\alpha }}^{\psi }{F_{\psi }}^{\beta }+\frac{1}{4}{g}^{\alpha \beta }{F}_{\psi \tau }{F}^{\psi \tau }\right)}$$

is used, then the Einstein field equations are called the Einstein–Maxwell equations (with cosmological constant Λ, taken to be zero in conventional relativity theory):

$${\displaystyle G^{\alpha \beta }+\mathrm{\Lambda }{g}^{\alpha \beta }=\frac{\kappa }{{\mu }_0}\left({F^{\alpha }}^{\psi }{F_{\psi }}^{\beta }+\frac{1}{4}{g}^{\alpha \beta }{F}_{\psi \tau }{F}^{\psi \tau }\right).}$$

Additionally, the covariant Maxwell equations are also applicable in free space:

$${\displaystyle \begin{array}{rl}{F^{\alpha \beta }}_{;\beta }& =0\\ F_{[\alpha \beta ;\gamma ]}& =\frac{1}{3}\left(F_{\alpha \beta ;\gamma }+F_{\beta \gamma ;\alpha }+F_{\gamma \alpha ;\beta }\right)=\frac{1}{3}\left(F_{\alpha \beta ,\gamma }+F_{\beta \gamma ,\alpha }+F_{\gamma \alpha ,\beta }\right)=0.\end{array}}$$

where the semicolon represents a covariant derivative, and the brackets denote anti-symmetrization. The first equation asserts that the 4-divergence of the 2-form F is zero, and the second that its exterior derivative is zero. From the latter, it follows by the Poincaré lemma that in a coordinate chart it is possible to introduce an electromagnetic field potential A_α such that

$${\displaystyle F_{\alpha \beta }=A_{\alpha ;\beta }-A_{\beta ;\alpha }=A_{\alpha ,\beta }-A_{\beta ,\alpha }}$$

in which the comma denotes a partial derivative. This is often taken as equivalent to the covariant Maxwell equation from which it is derived. However, there are global solutions of the equation that may lack a globally defined potential.

Solutions

Main article: Solutions of the Einstein field equations

The solutions of the Einstein field equations are metrics of spacetime. These metrics describe the structure of the spacetime including the inertial motion of objects in the spacetime. As the field equations are non-linear, they cannot always be completely solved (i.e. without making approximations). For example, there is no known complete solution for a spacetime with two massive bodies in it (which is a theoretical model of a binary star system, for example). However, approximations are usually made in these cases. These are commonly referred to as post-Newtonian approximations. Even so, there are several cases where the field equations have been solved completely, and those are called exact solutions.

The study of exact solutions of Einstein's field equations is one of the activities of cosmology. It leads to the prediction of black holes and to different models of evolution of the universe.

One can also discover new solutions of the Einstein field equations via the method of orthonormal frames as pioneered by Ellis and MacCallum. In this approach, the Einstein field equations are reduced to a set of coupled, nonlinear, ordinary differential equations. As discussed by Hsu and Wainwright, self-similar solutions to the Einstein field equations are fixed points of the resulting dynamical system. New solutions have been discovered using these methods by LeBlanc and Kohli and Haslam.

The linearized EFE

Main article: Linearized gravity

The nonlinearity of the EFE makes finding exact solutions difficult. One way of solving the field equations is to make an approximation, namely, that far from the source(s) of gravitating matter, the gravitational field is very weak and the spacetime approximates that of Minkowski space. The metric is then written as the sum of the Minkowski metric and a term representing the deviation of the true metric from the Minkowski metric, ignoring higher-power terms. This linearization procedure can be used to investigate the phenomena of gravitational radiation.

Polynomial form

Despite the EFE as written containing the inverse of the metric tensor, they can be arranged in a form that contains the metric tensor in polynomial form and without its inverse. First, the determinant of the metric in 4 dimensions can be written

$${\displaystyle det(g)=\frac{1}{24}{\epsilon }^{\alpha \beta \gamma \delta }{\epsilon }^{\kappa \lambda \mu \nu }{g}_{\alpha \kappa }{g}_{\beta \lambda }{g}_{\gamma \mu }{g}_{\delta \nu }}$$

using the Levi-Civita symbol; and the inverse of the metric in 4 dimensions can be written as:

$${\displaystyle g^{\alpha \kappa }=\frac{\frac{1}{6}{\epsilon }^{\alpha \beta \gamma \delta }{\epsilon }^{\kappa \lambda \mu \nu }{g}_{\beta \lambda }{g}_{\gamma \mu }{g}_{\delta \nu }}{det(g)}.}$$

Substituting this definition of the inverse of the metric into the equations then multiplying both sides by a suitable power of det(g) to eliminate it from the denominator results in polynomial equations in the metric tensor and its first and second derivatives. The action from which the equations are derived can also be written in polynomial form by suitable redefinitions of the fields.