Carbon emission trading

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

Allowance prices for carbon emission trade in all major emission trading schemes in Euro per ton of CO₂ emitted (from 2008 until August 2024)

Carbon emission trading (also called carbon market, emission trading scheme (ETS) or cap and trade) is a type of emissions trading scheme designed for carbon dioxide (CO₂) and other greenhouse gases (GHGs). A form of carbon pricing, its purpose is to limit climate change by creating a market with limited allowances for emissions. Carbon emissions trading is a common method that countries use to attempt to meet their pledges under the Paris Agreement, with schemes operational in China, the European Union, and other countries.

Emissions trading sets a quantitative total limit on the emissions produced by all participating emitters, which correspondingly determines the prices of emissions. Under emission trading, a polluter having more emissions than their quota has to purchase the right to emit more from emitters with fewer emissions. This can reduce the competitiveness of fossil fuels, which are the main driver of climate change. Instead, carbon emissions trading may accelerate investments into renewable energy, such as wind power and solar power.

However, such schemes are usually not harmonized with defined carbon budgets that are required to maintain global warming below the critical thresholds of 1.5 °C or "well below" 2 °C, with oversupply leading to low prices of allowances with almost no effect on fossil fuel combustion. Emission trade allowances currently cover a wide price range from €7 per tonne of CO₂ in China's national carbon trading scheme to €63 per tonne of CO₂ in the EU-ETS (as of September 2021).

Other greenhouse gases can also be traded but are quoted as standard multiples of carbon dioxide with respect to their global warming potential.

An international coalition to create a global carbon market, including a global, gradually declining, cap on emissions began to form in COP30. It can speed up emissions reduction seven-fold in all participating countries, while delivering $200 billion per year for clean-energy and social programs.

Purpose

The economic problem with climate change is that the emitters of greenhouse gases (GHGs) do not face the external costs of their actions, which include the present and future welfare of people, the natural environment, and the social cost of carbon. This can be addressed with the dynamic price model of emissions trading.

An emissions trading scheme for greenhouse gas emissions (GHGs) works by establishing property rights for the atmosphere. The atmosphere is a global public good, and GHG emissions are an international externality. In the cap-and-trade variant of emissions trading, a cap on access to a resource is defined and then allocated among users in the form of permits. Compliance is established by comparing actual emissions with permits surrendered. The setting of the cap affects the environmental integrity of carbon trading, and can result in both positive and negative environmental effects.

Emissions trading programmes such as the European Union Emissions Trading System (EU-ETS) complement the country-to-country trading stipulated in the Kyoto Protocol by allowing private trading of permits, coordinating with national emissions targets provided under the Kyoto Protocol. Under such programmes, a national or international authority allocates permits to individual companies based on established criteria, with a view to meeting targets at the lowest overall economic cost.

History

See also: Emissions trading § History

"Economy-wide pricing of carbon is the centre piece of any policy designed to reduce emissions at the lowest possible costs".

— Ross Garnaut, lead author of the Garnaut Climate Change Review in 2011

Carbon emission trading began in Rio de Janeiro in 1992, when 160 countries agreed the UN Framework Convention on Climate Change (UNFCCC). The necessary detail was left to be settled by the UN Conference of Parties (COP).

In 1997, the Kyoto Protocol was the first major agreement to reduce greenhouse gases. 38 developed countries committed themselves to targets and timetables. The resulting inflexible limitations on GHG growth could entail substantial costs if countries have to solely rely on their own domestic measures.

Carbon emissions trading increased rapidly in 2021 with the start of the Chinese national carbon trading scheme. The increasing costs of permits on the EU ETS have had the effect of increasing costs of coal power.

A 2019 study by the American Council for an Energy Efficient Economy finds that efforts to put a price on greenhouse gas emissions are growing in North America. In 2021, shipowners said they were against being included in the EU ETS.

Global Carbon Market Statistics

The global carbon market has experienced significant growth in recent years. In 2023, the value of the global carbon market reached a record high of 881 billion euros (approximately $949 billion), representing a 2% increase from the previous year. The European Union Emissions Trading System (EU ETS) remains the largest carbon market based on value, accounting for approximately 87% of the global market size in 2023.

In terms of trading volume, approximately 12.5 billion metric tons of carbon dioxide (GtCO₂) were traded in global carbon markets in 2022, which represented a decline of over 20% from the previous year but still an 18.2% increase compared to 2019 levels. Europe dominated the carbon trading volume, accounting for roughly 74% of the traded volume of CO₂ worldwide in 2022.

Economic aspects and tools

See also: Emissions trading § Economics, Carbon price, and Economic analysis of climate change

Economists generally agree that to regulate emissions efficiently, all polluters need to face the full marginal social costs of their actions. Regulation of emissions applied only to one economic sector or region drastically reduces the efficiency of efforts to reduce global emissions. There is, however, no scientific consensus over how to share the costs and benefits of reducing future climate change, or the costs and benefits of adapting to any future climate change.

Carbon offsets and credits

Renewable energy projects, such as these wind turbines near Aalborg, Denmark, constitute one common type of carbon offset project.

A carbon credit is a tradable instrument (typically a virtual certificate) that conveys a claim to have avoided greenhouse gas (GHG) emissions or to have enhanced removal of GHG from the atmosphere. One carbon credit represents the avoided or enhanced removal of one metric ton of carbon dioxide or its carbon dioxide-equivalent (CO₂e).

Carbon offsetting is the practice of using carbon credits to offset or counter an entity's greenhouse gas inventory emissions in line with reporting programs or institutional emissions targets/goals. Carbon credit trading mechanisms (i.e., crediting programs), enable project developers to implement projects that mitigate GHGs and receive carbon credits which can be sold to interested buyers who may use the credits to claim they have offset their inventory GHG emissions. Similar to "offsetting", carbon credits that are permitted as compliance instruments within regulatory compliance markets (e.g., The European Union Emission Trading Scheme or the California Cap-n-Trade program) can be used by regulated entities to report lower emissions and achieve compliance status (with limitations around their use that vary by compliance program). Aside from "offsetting", carbon credits can also be used to make contributions toward global net zero GHG-level targets. It is an individual buyer's choice how to use, or "retire", the carbon credit.

Projects entail mitigation actions that avoid or enhance the removal of GHG emissions. Projects are implemented in line with the standards of crediting programs, including their methodologies, rules, and requirements. Methodologies are approved for each specific project type (e.g., tree planting, mangrove restoration, early retirement of coal powerplants). Provided a project fulfills all of the requirements and provisions of a crediting program, it will be issued credits that can be sold to buyers. Each crediting program typically has its own carbon credit 'label' such as CDM's Certified Emission Reductions (CERs), Article 6.4 Mechanism Emission Reductions (A6.4ERs), VCS' Verified Emission Reductions (VERs), ACR's Emission Reduction Tonnes, Climate Action Reserves' Climate Reserve Tonnes (CRTs), etc.

Hundreds of GHG mitigation project types exist and have approved methodologies with established crediting programs. The program that defined the first phase of carbon market development, the Clean Development Mechanism (CDM) provides a summary booklet of its many approved methodologies. But each crediting program has its own list of approved methodologies, for example unless explicitly stated, an ACR approved methodology could not be used by someone trying to work through Verra's Verified Carbon Standard. Carbon credits are a form of carbon pricing, along with carbon taxes, and Carbon Border Adjustment Mechanisms (CBAM). Carbon credits are intended to be fungible across different markets, but some compliance markets and reporting programs limit eligibility to specified carbon credit types or characteristics (e.g., vintage, project origin, project type).

Carbon leakage

A domestic carbon emissions trading scheme is constrained in its regulatory jurisdiction. GHG emissions may thus leak to another region or sector with less regulation. Generally, leakages reduce the effectiveness of domestic emission abatement efforts. Notwithstanding, leakages may also be negative in nature, increasing the effectiveness of domestic abatement efforts. For example, a carbon tax applied only to developed countries might lead to a positive leakage to developing countries. However, a negative leakage might also occur due to technological developments driven by domestic regulation of GHGs, helping to reduce emissions even in less regulated regions.

The current state of carbon emissions trading shows that roughly 22% of global greenhouse emissions are covered by 64 carbon taxes and emission trading systems as of 2021. Energy intensive industries that are covered by such instruments may view the regulatory disparity between jurisdictions as a loss of competitiveness. They may therefore make strategic production decisions that involve carbon leakage. To mitigate carbon leakage and its effects on the environment, policymakers need to harmonize international climate policies and provide incentives to prevent companies from relocating production to regions with more lenient environmental regulations.

Free emission permits, given to sectors vulnerable to international competition, are one way of addressing carbon leakage by acting as a subsidy for the sector in question. The Garnaut Climate Change Review considered the free allocation of permits unjustified in any circumstances, arguing that governments could deal with market failure or claims for compensation more transparently with the revenue from full auctioning of permits.

Border Adjustment

Further information: Eco-tariff

Another economically efficient solution to carbon leakage is border adjustment, where tariffs are set on imported goods from less regulated countries. A problem with border adjustments is that they might be used as a disguise for trade protectionism. Some types of border adjustment may also not prevent emissions leakage. The EU Carbon Border Adjustment Mechanism takes in effect for 6 sectors in 2026.

A study examining the potential effects of the European Union's Carbon Border Adjustment Mechanism (CBAM) suggests that the regulation may impose additional costs on EU companies. The analysis focused on publicly listed firms operating in CBAM-covered sectors across 75 countries and assessed stock market reactions at three key stages of the EU legislative process. The findings indicate that, around the relevant announcement dates, EU-based firms experienced larger declines in share prices than non-EU firms in the same sectors, with differences in returns ranging between 2 and 3 percentage points.

The study finds heterogeneity among EU firms. Companies with a greater reliance on suppliers located outside the EU showed more pronounced negative market reactions than those with fewer non-EU suppliers. In addition, the adverse response was concentrated among firms with relatively low profit margins, often used as an indicator of limited market power. The results suggest that some EU firms may face higher import costs either because of their dependence on non-EU suppliers or because limited pricing power constrains their ability to pass on carbon-related costs to foreign exporters subject to the CBAM.

Relevance to climate justice

Carbon trading can be helpful to achieve climate justice. It can transfer money from rich countries, which tend to have higher emissions, to countries with lower incomes and lower emissions for improved climate action.

Cap-and-trade systems have also been linked to causing environmental justice as low-income communities receive less benefits from reduced emissions and are often located near the emitters. Companies under emission trading systems will often emit more pollutants not covered by the system and disproportionately affect low-income communities.

Potential global carbon market

The Paris Agreement provided a legal base for the creation of a global carbon market, which has a potentially significant role in stopping climate change. In the beginning of 2024, the idea made some progress with the Bonn meeting where new tools and supervisory bodies were created.

The rules of the European Union Emissions Trading System include the possibility of connecting it with other trading systems. This has already happened with the Switzerland emissions trading system China expressed a support for a global carbon market, saying it is better than the EU Carbon Border Adjustment Mechanism.

In 2023 the global value of carbon markets was $948.75 billion. It is expected to reach 2.68 trillion dollars by 2028  and 22 trillion by 2050.

Merging the ETC of China and the EU can be something that sends "a powerful signal to the rest of the world and catalyzes international buy-in" while strongly increasing the efficiency of the system and allowing both countries to attain higher results with less spending.

A global carbon market can speed up emissions reduction seven-fold in all participating countries, while delivering $200 billion per year for clean-energy and social programs. An international coalition for creating it called Open Coalition on Compliance Carbon Markets began to form in COP 30. The plan is to create a global emissions cap beginning with a level close to current emissions rate, and then reducing it until reaching net-zero by 2050. For any activity which causes emissions, people would buy allowances. As the cap decreases, the cost of the allowances will increase, creating an incentive for decarbonization. There will be a border adjustment mechanism governed by all participants. Poorer countries can not pay or pay less and part of the revenue will be spent on helping them address the climate crisis. The formal launch of the Coalition is expected during 2026.

In voluntary carbon markets, the Integrity Council for the Voluntary Carbon Market publishes the Core Carbon Principles (CCPs) and related guidance intended to define high-level integrity criteria for carbon credits.

Allocation of permits

Tradable emissions permits can be issued to firms within an ETS by two main ways: by free allocation of permits to existing emitters or by auction. In the first case, the government receives no carbon revenue. In the second it receives the full value of the permits, on average. In either case, permits will be equally scarce and just as valuable to market participants, such that the price at sale will be the same in either case.

Generally, emitters will profit from permits allocated to them for free. But if they must pay, their profits will be reduced. If the carbon price equals the true social cost of carbon, then long-run profit reduction will reflect the consequences of paying this new cost. If having to pay this cost is unexpected, then there will likely be a one-time loss due to the change in regulations and not simply due to paying the real cost of carbon. However, if there is advanced notice of this change, or if the carbon price is introduced gradually, this one-time regulatory cost will be minimized. There has now been enough advance notice of carbon pricing that this effect should be negligible on average.

Grandfathering

Allocating permits based on past emissions is called "grandfathering". Grandfathering permits can lead to perverse incentives, such as a firm being given fewer permits in the future for aiming to cut emissions drastically. Another method of grandfathering is to base allocations on current production of economic goods rather than historical emissions. Under this method of allocation, the government will set a benchmark level of emissions for each good deemed to be sufficiently trade exposed and allocate firms units based on their production of this good. However, allocating permits in proportion to output implicitly subsidises production.

The Garnaut Climate Change Review noted that grandfathered permits are not free of cost. As the permits are scarce, they have value, and the benefit of that value is acquired in full by the emitter. The cost is imposed elsewhere in the economy, typically on consumers who cannot pass on the costs: The cost of a grandfathered permit may be regarded as the opportunity cost of not selling the permit at full value. As a result, profit-maximising firms receiving free permits will raise prices to customers because of the new, non-zero cost of emissions. This gives permit-liable polluters an incentive to reduce their emissions. However, if a firm sells the same amount of output as before that cap, with no change in production technology, the full value of permits received for free becomes windfall profits. However, since the cap reduces output and often causes the company to incur costs to increase efficiency, windfall profits will be less than the full value of its free permits.

Grandfathering may also slow down technological development towards less polluting technologies. The Garnaut Report noted that any method for free permit allocation will have the disadvantages of high complexity, high transaction costs, value-based judgements, and the use of arbitrary emissions baselines. Garnaut also noted that the complexity of free allocation and the large amounts of money involved encourage non-productive rent-seeking behaviour and lobbying of governments — activities that dissipate economic value.

At the same time, allocating permits can be used as a measure to protect domestic firms who are internationally exposed to competition. This happens when domestic firms compete against other firms that are not subject to the same regulation. This argument in favor of allocation of permits has been used in the EU ETS, where industries that have been judged to be internationally exposed have been given permits for free.

The International Air Transport Association, whose 230 member airlines comprise 93% of all international traffic, argue that emissions levels should be based on industry averages rather than using individual companies' previous emissions levels to set their future permit allowances, stating that "would penalise airlines that took early action to modernise their fleets, while a benchmarking approach, if designed properly, would reward more efficient operations".

Auctioning

Hepburn et al. state that, empirically, businesses tend to oppose auctioning of emissions permits, while economists almost uniformly recommend auctioning permits. Auctioning permits provides the government with revenues, which can be used to fund low-carbon investment and cuts in distortionary taxes. Auctioning permits can therefore be more efficient and equitable than allocating permits. Garnaut stated that full auctioning will provide greater transparency and accountability and lower implementation and transaction costs as governments retain control over the permit revenue. Auctions of units are more flexible in distributing costs, provide more incentives for innovation, lessen the political arguments over the allocation of economic rents, and reduce tax distortions. Recycling of revenue from permit auctions could also offset a significant proportion of the economy-wide social costs of a cap and trade scheme.

The perverse incentive of grandfathering can be alleviated through auctioning.

Permit supply level

Regulatory agencies run the risk of issuing too many emission credits, which can result in a very low price on emission permits. This reduces the incentive that permit-liable firms have to cut back their emissions. On the other hand, issuing too few permits can result in an excessively high permit price. An argument has been made for a hybrid instrument having a price floor and a price ceiling. However, a price-ceiling safety value removes the certainty of a particular quantity limit of emissions.

Emissions trading and carbon taxes around the world (2024)

  ETS and carbon tax implemented

  ETS implemented

  Carbon tax implemented

  ETS or carbon tax under consideration or under development

Criticisms

See also: Criticism of the Kyoto Protocol § Criticism of Carbon Trade

Chicago Climate Justice activists protesting cap and trade legislation in front of Chicago Climate Exchange building in Chicago Loop

Emissions trading has been criticized for a variety of reasons. For one, it has been argued that climate change requires more radical solutions than pollution trading schemes, and that systemic changes must be made to reduce fossil fuel usage. At the same time, carbon credits have been seen as enabling large companies to pollute the environment at the expense of local communities. Carbon trading has also been criticised as a form of colonialism, in which rich countries maintain their levels of consumption while getting credit for carbon savings in inefficient industrial projects.

Groups such as the Corner House have argued that the market will choose the easiest means to save a given quantity of carbon in the short term, which may be different from the means to reduce climate change. In September 2010, campaigning group FERN released "Trading Carbon: How it works and why it is controversial"^[ which compiles many of the arguments against carbon trading. According to Carbon Trade Watch, carbon trading has had a "disastrous track record". The effectiveness of the EU ETS was criticized, and it was argued that the CDM had routinely favoured "environmentally ineffective and socially unjust projects".

Some groups have claimed that non-existent emission reductions can be recorded under the Kyoto Protocol due to the surplus of allowances that some countries possess. For example, Russia had a surplus of allowances due to its economic collapse following the end of the Soviet Union. Other countries could have bought these allowances from Russia, but this would not have reduced emissions. In practice, as of 2010, Kyoto Parties had not yet chosen not to buy these surplus allowances.

The complexity of cap and trade schemes around the world has resulted in the uncertainties around such schemes in Australia, Canada, China, the EU, India, Japan, New Zealand, and the US. As a result, some organizations have had little incentive to innovate and comply, resulting in an ongoing battle of stakeholder contestation for the past two decades.

Proposals for alternative schemes to avoid the problems of cap-and-trade schemes include Cap and Share, which was considered by the Irish Parliament in 2008, and the Sky Trust schemes.

Carbon emission trading without border adjustments for exports leads to reduced global competitiveness for carbon-intensive products.

Some critics in the EU blamed the EU ETS for contributing to the 2021 global energy crisis.^[95][96] In August 2022, Polish Prime Minister Mateusz Morawiecki called for a temporary suspension of the EU ETS to stabilize electricity prices, saying the "price increase [on the ETS] is out of control and hitting the household budgets of EU citizens."

Abuses

The Financial Times published an article about cap-and-trade systems, which argued that "Carbon markets create a muddle" and "...leave much room for unverifiable manipulation". Emissions trading schemes have also been criticised for the potential of creating a new speculative market through the commodification of environmental risks through financial derivatives.

Annie Leonard's 2009 documentary The Story of Cap and Trade criticized carbon emissions trading for the free permits to major polluters giving them unjust advantages, cheating in connection with carbon offsets, and as a distraction from the search for other solutions.

In China, some companies started artificial production of greenhouse gases with sole purpose of recycling and gaining carbon credits. Similar practices happened in India. Earned credit were then sold to companies in US and Europe.

Corporate and governmental carbon emission trading schemes have been modified in ways that have been attributed to permitting money laundering to take place.

Examples by country

Australia

Main articles: Garnaut Climate Change Review, Carbon Pollution Reduction Scheme, and Clean Energy Bill 2011

In 2003 the New South Wales (NSW) state government unilaterally established the New South Wales Greenhouse Gas Abatement Scheme^[105] to reduce emissions by requiring electricity generators and large consumers to purchase NSW Greenhouse Abatement Certificates (NGACs). This has prompted the rollout of free energy-efficient compact fluorescent lightbulbs and other energy-efficiency measures, funded by the credits. This scheme has been criticised by the Centre for Energy and Environmental Markets (CEEM) of the University of New South Wales (UNSW) because of its lack of effectiveness in reducing emissions, its lack of transparency and its lack of verification of the additionality of emission reductions.

Prior to the 2007 federal election, both the incumbent Howard Coalition government and the Rudd Labor opposition promised to implement an emissions trading scheme (ETS). Labor won the election, and the new government proceeded to implement an ETS. The new Rudd government introduced the Carbon Pollution Reduction Scheme, which the Liberal Party of Australia (now led by Malcolm Turnbull) supported. Tony Abbott questioned an ETS, advocating a "simple tax" as the best way to reduce emissions. Shortly before the carbon vote, Abbott defeated Turnbull in a leadership challenge (December 1, 2009), and from there on the Liberals opposed the ETS. This left the Rudd Labor government unable to secure passage of the bill, and it was subsequently withdrawn.

Julia Gillard defeated Rudd in a leadership challenge, becoming Federal Prime Minister in June 2010. She promised that she would not introduce a carbon tax, but would look to legislate a price on carbon when taking the government to the 2010 election. In the first Australian hung-parliament result in 70 years, the Gillard Labor government required the support of crossbenchers - including the Greens. One requirement for Greens' support was a carbon price, which Gillard proceeded with in forming a minority government. A fixed carbon-price would proceed to a floating-price ETS within a few years under the plan. The fixed price lent itself to characterisation as a "carbon tax", and when the government proposed the Clean Energy Bill in February 2011, the opposition denounced it as a broken election promise.

The Lower House passed the bill in October 2011 and the Upper House in November 2011. The Liberal Party vowed to repeal the bill if elected. The bill thus resulted in passage of the Clean Energy Act, which possessed a great deal of flexibility in its design and uncertainty over its future.

The Liberal/National coalition government elected in September 2013 promised to reverse the climate legislation of the previous government. In July 2014, the carbon tax was repealed - as well as the Emissions Trading Scheme (ETS) that was to start in 2015.

Canada

The Canadian provinces of Quebec and Nova Scotia operate an emissions trading scheme. Quebec links its program with the US state of California through the Western Climate Initiative.

China

Main article: Chinese national carbon trading scheme

The Chinese national carbon trading scheme is the largest in the world. It is an intensity-based trading system for carbon dioxide emissions by China, which started operating in 2021. The initial design of the system targets a scope of 3.5 billion tons of carbon dioxide emissions that come from 1700 installations. It has made a voluntary pledge under the UNFCCC to lower CO₂ per unit of GDP by 40–45% in 2020 when comparing to the 2005 levels.

In November 2011, China approved pilot tests of carbon trading in seven provinces and cities—Beijing, Chongqing, Shanghai, Shenzhen, Tianjin, as well as Guangdong Province and Hubei Province, with different prices in each region. The pilot is intended to test the waters and provide valuable lessons for the design of a national system in the near future. Their successes or failures will, therefore, have far-reaching implications for carbon market development in China in terms of trust in a national carbon trading market. Some of the pilot regions can start trading as early as 2013/2014. National trading is expected to start in 2017, latest in 2020.

The effort to start a national trading system has faced some problems that took longer than expected to solve, mainly in the complicated process of initial data collection to determine the base level of pollution emission. According to the initial design, there will be eight sectors that are first included in the trading system: chemicals, petrochemicals, iron and steel, non-ferrous metals, building materials, paper, power and aviation, but many of the companies involved lacked consistent data. Therefore, by the end of 2017, the allocation of emission quotas have started but it has been limited to only the power sector and will gradually expand, although the operation of the market is yet to begin. In this system, Companies that are involved will be asked to meet target level of reduction and the level will contract gradually.

An overview of studies related to the impacts of the ETS found that it "generates clear environmental and social benefits but exhibits mixed economic and innovation effects." The system reduce carbon emissions by up to 18.2%, reducing pollution, inequality and improve welfare so that 33% of all health benefits from air pollution control policies in China are the benefits from the ETS. Some researchers found it can reduce GDP or inhibit innovation. By 2027, it is expected to expand to all major industrial emmiters and began to establish an absolute cap.

European Union

Price of CO₂ in the EU Emissions Trading System

The European Union Emissions Trading System (EU ETS) is a carbon emission trading scheme (or cap and trade scheme) that began in 2005 and is intended to lower greenhouse gas emissions in the EU. Cap and trade schemes limit emissions of specified pollutants over an area and make polluters pay for their pollution, requiring them to buy allowances to emit enough to cover their emissions, from the EU or from other companies. The money is channeled to environmental and social goals. As of 2026 the ETS covers around 40% of the EU's greenhouse gas emissions. The cap decline gradually and should reach zero by 2039. After this year no more allowances will be distributed and when the unused allowances will end, no more emissions will be permitted.

As from 2027 road transport, buildings and industrial installation that fell out of EU ETS will be covered by a new EU ETS2. The "old" ETS and the new EU ETS2 allowances will be traded independently. A major difference to the ETS is that ETS2 will cover the CO₂ emissions upstream - fuel suppliers rather than consumers will be obliged to cover emissions with ETS2 emission allowances. The 2 systems will cover 75% of the GHG emissions of the European Union.

Compared to 2005, when the EU ETS was first implemented, the proposed caps for 2020 represent a 21% reduction in greenhouse gases. This target was achieved six years early as emissions in the ETS fell to 1.812 billion (10⁹) tonnes in 2014.

During the years 2005–2025, GHG emissions in the sectors covered by the ETS declined by around 50%, declining only by 20% in the not covered sectors.  A 2020 study showed that between 2008 and 2016 the ETS reduced CO₂ emissions by 11.5% in covered sectors despite low carbon price. A 2024 study estimate the emission reduction effect at 7%. According to a 2023 study the ETS, reduced emissions by 10% between 2005 and 2012 with no impacts on profits or employment for regulated firms. A 2024 study demonstrated that the ETS has contributed to reduce atmospheric levels of air pollutants in the EU including sulfur dioxide, fine particulate matter, and nitrogen oxide. This reduction has translated in local health co-benefits, alongside the system's primary goal of mitigating climate change. EU countries view the emissions trading scheme as necessary for meeting climate goals. A strong carbon market guides investors and industry in their transition from fossil fuels.

India

Trading is set to begin in 2014 after a three-year rollout period. It is a mandatory energy efficiency trading scheme covering eight sectors responsible for 54 per cent of India's industrial energy consumption. India has pledged a 20 to 25 per cent reduction in emission intensity from 2005 levels by 2020. Under the scheme, annual efficiency targets will be allocated to firms. Tradable energy-saving permits will be issued depending on the amount of energy saved during a target year.

Japan

Japan as a country does not have a compulsory emissions trading scheme. The government in 2010 (the Hatoyama cabinet) had planned to introduce one, but the plan lost momentum after Hatoyama resigned as prime minister, due partly from industrial opposition, and was eventually shelved. Japan has a voluntary scheme. Furthermore, the Kyoto Prefecture has a voluntary emissions trading scheme.

Two regional mandatory schemes exist however, in Tokyo and Saitama Prefecture. The city of Tokyo consumes as much energy as "entire countries in Northern Europe, and its production matches the GNP of the world's 16th largest country". A cap-and-trade carbon trading scheme launched in April 2010 covers the top 1,400 emitters in Tokyo, and is enforced and overseen by the Tokyo Metropolitan Government.^[140] Phase 1, which was similar to Japan's voluntary scheme, ran until 2015. Emitters had to cut their emissions by 6% or 8% depending on the type of organization; from 2011, those who exceed their limits were required to buy matching allowances, or invest in renewable-energy certificates, or offset credits issued by smaller businesses or branch offices. Polluters that failed to comply were liable up to 500,000 yen in fines plus credits for 1.3 times excess emissions. In its fourth year, emissions were reduced by 23% compared to base-year emissions. In phase 2 (FY2015–FY2019), the target was expected to increase to 15–17%. The aim was to cut Tokyo's carbon emissions by 25% from 2000 levels by 2020.

One year after Tokyo launched its cap-and-trade scheme, the neighbouring Saitama Prefecture launched a highly similar scheme. The two schemes are connected.

New Zealand

New Zealand Unit Prices

The New Zealand Emissions Trading Scheme (NZ ETS) is an all-gases partial-coverage uncapped domestic emissions trading scheme that features price floors, forestry offsetting, free allocation and auctioning of emissions units.

The NZ ETS was first legislated in the Climate Change Response (Emissions Trading) Amendment Act 2008 in September 2008 under the Fifth Labour Government of New Zealand and then amended in November 2009 and in November 2012 by the Fifth National Government of New Zealand.

The NZ ETS was until 2015 highly linked to international carbon markets as it allowed unlimited importing of most of the Kyoto Protocol emission units. There is a domestic emission unit; the 'New Zealand Unit' (NZU), which was initially issued by free allocation to emitters until auctions of units commenced in 2020. The NZU is equivalent to 1 tonne of carbon dioxide. Free allocation of units varies between sectors. The commercial fishery sector (who are not participants) received a one-off free allocation of units on a historic basis. Owners of pre-1990 forests received a fixed free allocation of units. Free allocation to emissions-intensive industry, is provided on an output-intensity basis. For this sector, there is no set limit on the number of units that may be allocated. The number of units allocated to eligible emitters is based on the average emissions per unit of output within a defined 'activity'. Bertram and Terry (2010, p 16) state that as the NZ ETS does not 'cap' emissions, the NZ ETS is not a cap and trade scheme as understood in the economics literature.

Some stakeholders have criticised the New Zealand Emissions Trading Scheme for its generous free allocations of emission units and the lack of a carbon price signal (the Parliamentary Commissioner for the Environment), and for being ineffective in reducing emissions (Greenpeace Aotearoa New Zealand).

South Korea

South Korea's national emissions trading scheme officially launched on January 1, 2015, covering 525 entities from 23 sectors. With a three-year cap of 1.8687 billion tCO₂e, it now forms the second-largest carbon market in the world, following the EU ETS. This amounts to roughly two-thirds of the country's emissions. The Korean emissions trading scheme is part of the Republic of Korea's efforts to reduce greenhouse gas emissions by 30% compared to the business-as-usual scenario by 2020.

United Kingdom

The UK Emissions Trading Scheme (UK ETS) is the carbon emission trading scheme of the United Kingdom. It is cap and trade and came into operation on 1 January 2021 following the UK's departure from the European Union. The cap is reduced in line with the UK's 2050 net zero commitment.

Carbon Price Support (CPS) is an additional tax, paid by electricity generation companies that use fossil fuels, introduced in 2013 in response to the low prices then on the European Union Emissions Trading System.

United States

See also: Climate change in the United States

The American Clean Energy and Security Act (H.R. 2454), a greenhouse gas cap-and-trade bill, was passed on June 26, 2009, in the House of Representatives by a vote of 219–212. The bill originated in the House Energy and Commerce Committee. It was introduced by Representatives Henry A. Waxman and Edward J. Markey. The political advocacy organizations FreedomWorks and Americans for Prosperity, funded by brothers David and Charles Koch of Koch Industries, encouraged the Tea Party movement to focus on defeating the legislation. Although cap and trade also gained a significant foothold in the Senate via the efforts of Republican Lindsey Graham, Independent and former Democrat Joe Lieberman, and Democrat John Kerry, the legislation died in the Senate.

President Barack Obama's proposed 2010 United States federal budget wanted to support clean energy development with a 10-year investment of US$15 billion per year, generated from the sale of greenhouse gas emissions credits. Under the proposed cap-and-trade program, all GHG emissions credits would have been auctioned off, generating an estimated $78.7 billion in additional revenue in FY 2012, steadily increasing to $83 billion by FY 2019. The proposal was never made law. Failing to get congressional approval for such a scheme, President Barack Obama instead acted through the United States Environmental Protection Agency to attempt to adopt the Clean Power Plan, which does not feature emissions trading. The plan was subsequently challenged by the administration of President Donald Trump.

In 2006, the California State Legislature adopted the California Assembly Bill 32 (AB32), the Global Warming Solutions Act that let to a statewide cap-and-trade program that began in 2012. California and Quebec linked their cap-and-trade programs in 2014, sharing one carbon market.

In 2021, Washington state instituted its own emissions trading system, which it called "Cap-and-Invest." Revenue from the auctioning of carbon allowances is directly invested in programs intended to address climate change.

In the United States, most polling shows large support for emissions trading.

Inertial frame of reference

From Wikipedia, the free encyclopedia

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

In classical physics and special relativity, an inertial frame of reference (also called an inertial space or a Galilean reference frame) is a frame of reference in which objects exhibit inertia: they remain at rest or in uniform motion relative to the frame until acted upon by external forces. In such a frame, the laws of nature can be observed without the need to correct for acceleration.

All frames of reference with zero acceleration are in a state of constant rectilinear motion (straight-line motion) with respect to one another. In such a frame, an object with zero net force acting on it, is perceived to move with a constant velocity, or, equivalently, Newton's first law of motion holds. Such frames are known as inertial. Some physicists, like Isaac Newton, originally thought that one of these frames was absolute — the one approximated by the fixed stars. However, this is not required for the definition, and it is now known that those stars are in fact moving, relative to one another.

According to the principle of special relativity, all physical laws look the same in all inertial reference frames, and no inertial frame is privileged over another. Measurements of objects in one inertial frame can be converted to measurements in another by a simple transformation — the Galilean transformation in Newtonian physics or the Lorentz transformation (combined with a translation) in special relativity; these approximately match when the relative speed of the frames is low, but differ as it approaches the speed of light.

By contrast, a non-inertial reference frame is accelerating. In such a frame, the interactions between physical objects vary depending on the acceleration of that frame with respect to an inertial frame. Viewed from the perspective of classical mechanics and special relativity, the usual physical forces caused by the interaction of objects have to be supplemented by fictitious forces caused by inertia.Viewed from the perspective of general relativity theory, the fictitious (i.e. inertial) forces are attributed to geodesic motion in spacetime.

Due to Earth's rotation, its surface is not an inertial frame of reference. The Coriolis effect can deflect certain forms of motion as seen from Earth, and the centrifugal force will reduce the effective gravity at the equator. Nevertheless, for many applications the Earth is an adequate approximation of an inertial reference frame.

Introduction

The motion of a body can only be described relative to something else—other bodies, observers, or a set of spacetime coordinates. These are called frames of reference. According to the first postulate of special relativity, all physical laws take their simplest form in an inertial frame, and there exist multiple inertial frames interrelated by uniform translation:

Special principle of relativity: If a system of coordinates K is chosen so that, in relation to it, physical laws hold good in their simplest form, the same laws hold good in relation to any other system of coordinates K' moving in uniform translation relatively to K.

— Albert Einstein: The foundation of the general theory of relativity, Section A, §1

This simplicity manifests itself in that inertial frames have self-contained physics without the need for external causes, while physics in non-inertial frames has external causes. The principle of simplicity can be used within Newtonian physics as well as in special relativity:

The laws of Newtonian mechanics do not always hold in their simplest form...If, for instance, an observer is placed on a disc rotating relative to the earth, he/she will sense a 'force' pushing him/her toward the periphery of the disc, which is not caused by any interaction with other bodies. Here, the acceleration is not the consequence of the usual force, but of the so-called inertial force. Newton's laws hold in their simplest form only in a family of reference frames, called inertial frames. This fact represents the essence of the Galilean principle of relativity:
   The laws of mechanics have the same form in all inertial frames.

— Milutin Blagojević: Gravitation and Gauge Symmetries, p. 4

However, this definition of inertial frames is understood to apply in the Newtonian realm and ignores relativistic effects.

In practical terms, the equivalence of inertial reference frames means that scientists within a box moving with a constant absolute velocity cannot determine this velocity by any experiment. Otherwise, the differences would set up an absolute standard reference frame. According to this definition, supplemented with the constancy of the speed of light, inertial frames of reference transform among themselves according to the Poincaré group of symmetry transformations, of which the Lorentz transformations are a subgroup. In Newtonian mechanics, inertial frames of reference are related by the Galilean group of symmetries.

Newton's inertial frame of reference

Absolute space

Main article: Absolute space and time

Newton posited an absolute space considered well-approximated by a frame of reference stationary relative to the fixed stars. An inertial frame was then one in uniform translation relative to absolute space. However, some "relativists", even at the time of Newton, felt that absolute space was a defect of the formulation, and should be replaced.

The expression inertial frame of reference (German: Inertialsystem) was coined by Ludwig Lange in 1885, to replace Newton's definitions of "absolute space and time" with a more operational definition:

A reference frame in which a mass point thrown from the same point in three different (non co-planar) directions follows rectilinear paths each time it is thrown, is called an inertial frame.

The inadequacy of the notion of "absolute space" in Newtonian mechanics is spelled out by Blagojevich:

• The existence of absolute space contradicts the internal logic of classical mechanics since, according to the Galilean principle of relativity, none of the inertial frames can be singled out.
• Absolute space does not explain inertial forces since they are related to acceleration with respect to any one of the inertial frames.
• Absolute space acts on physical objects by inducing their resistance to acceleration but it cannot be acted upon.

— Milutin Blagojević: Gravitation and Gauge Symmetries, p. 5

The utility of operational definitions was carried much further in the special theory of relativity. Some historical background including Lange's definition is provided by DiSalle, who says in summary:

The original question, "relative to what frame of reference do the laws of motion hold?" is revealed to be wrongly posed. The laws of motion essentially determine a class of reference frames, and (in principle) a procedure for constructing them.

— Robert DiSalle Space and Time: Inertial Frames

Newtonian mechanics

Classical theories that use the Galilean transformation postulate the equivalence of all inertial reference frames. The Galilean transformation transforms coordinates from one inertial reference frame, ${\displaystyle \mathbf{s}}$, to another, ${\displaystyle {\mathbf{s}}^{\mathrm{\prime }}}$, by simple addition or subtraction of coordinates:

${\displaystyle {\mathbf{r}}^{\mathrm{\prime }}=\mathbf{r}-{\mathbf{r}}_0-\mathbf{v}t}$

${\displaystyle t^{\mathrm{\prime }}=t-t_0}$

where r₀ and t₀ represent shifts in the origin of space and time, and v is the relative velocity of the two inertial reference frames. Under Galilean transformations, the time t₂ − t₁ between two events is the same for all reference frames and the distance between two simultaneous events (or, equivalently, the length of any object, |r₂ − r₁|) is also the same.

Figure 1: Two frames of reference moving with relative velocity ${\displaystyle \stackrel{\overrightarrow{v}}{}}$. Frame S' has an arbitrary but fixed rotation with respect to frame S. They are both inertial frames provided a body not subject to forces appears to move in a straight line. If that motion is seen in one frame, it will also appear that way in the other.

Within the realm of Newtonian mechanics, an inertial frame of reference, or inertial reference frame, is one in which Newton's first law of motion is valid. However, the principle of special relativity generalizes the notion of an inertial frame to include all physical laws, not simply Newton's first law.

Newton viewed the first law as valid in any reference frame that is in uniform motion (neither rotating nor accelerating) relative to absolute space; as a practical matter, "absolute space" was considered to be the fixed stars In the theory of relativity the notion of absolute space or a privileged frame is abandoned, and an inertial frame in the field of classical mechanics is defined as:

An inertial frame of reference is one in which the motion of a particle not subject to forces is in a straight line at constant speed.

Hence, with respect to an inertial frame, an object or body accelerates only when a physical force is applied, and (following Newton's first law of motion), in the absence of a net force, a body at rest will remain at rest and a body in motion will continue to move uniformly—that is, in a straight line and at constant speed. Newtonian inertial frames transform among each other according to the Galilean group of symmetries.

If this rule is interpreted as saying that straight-line motion is an indication of zero net force, the rule does not identify inertial reference frames because straight-line motion can be observed in a variety of frames. If the rule is interpreted as defining an inertial frame, then being able to determine when zero net force is applied is crucial. The problem was summarized by Einstein:

The weakness of the principle of inertia lies in this, that it involves an argument in a circle: a mass moves without acceleration if it is sufficiently far from other bodies; we know that it is sufficiently far from other bodies only by the fact that it moves without acceleration.

— Albert Einstein: The Meaning of Relativity, p. 58

There are several approaches to this issue. One approach is to argue that all real forces drop off with distance from their sources in a known manner, so it is only needed that a body is far enough away from all sources to ensure that no force is present. A possible issue with this approach is the historically long-lived view that the distant universe might affect matters (Mach's principle). Another approach is to identify all real sources for real forces and account for them. A possible issue with this approach is the possibility of missing something, or accounting inappropriately for their influence, perhaps, again, due to Mach's principle and an incomplete understanding of the universe. A third approach is to look at the way the forces transform when shifting reference frames. Fictitious forces, those that arise due to the acceleration of a frame, disappear in inertial frames and have complicated rules of transformation in general cases. Based on the universality of physical law and the request for frames where the laws are most simply expressed, inertial frames are distinguished by the absence of such fictitious forces.

Newton enunciated a principle of relativity himself in one of his corollaries to the laws of motion:

The motions of bodies included in a given space are the same among themselves, whether that space is at rest or moves uniformly forward in a straight line.

— Isaac Newton: Principia, Corollary V, p. 88 in Andrew Motte translation

This principle differs from the special principle in two ways: first, it is restricted to mechanics, and second, it makes no mention of simplicity. It shares the special principle of the invariance of the form of the description among mutually translating reference frames. The role of fictitious forces in classifying reference frames is pursued further below.

Special relativity

Main article: Special relativity

Einstein's theory of special relativity, like Newtonian mechanics, postulates the equivalence of all inertial reference frames. However, because special relativity postulates that the speed of light in free space is invariant, the transformation between inertial frames is the Lorentz transformation, not the Galilean transformation which is used in Newtonian mechanics.

The invariance of the speed of light leads to counter-intuitive phenomena, such as time dilation, length contraction, and the relativity of simultaneity. The predictions of special relativity have been extensively verified experimentally. The Lorentz transformation reduces to the Galilean transformation as the speed of light approaches infinity or as the relative velocity between frames approaches zero.

Examples

Simple example

Figure 1: Two cars moving at different but constant velocities observed from stationary inertial frame S attached to the road and moving inertial frame S′ attached to the first car.

Consider a situation common in everyday life. Two cars travel along a road, both moving at constant velocities. See Figure 1. At some particular moment, they are separated by 200 meters. The car in front is traveling at 22 meters per second and the car behind is traveling at 30 meters per second. If we want to find out how long it will take the second car to catch up with the first, there are three obvious "frames of reference" that we could choose.

First, we could observe the two cars from the side of the road. We define our "frame of reference" S as follows. We stand on the side of the road and start a stop-clock at the exact moment that the second car passes us, which happens to be when they are a distance d = 200 m apart. Since neither of the cars is accelerating, we can determine their positions by the following formulas, where ${\displaystyle x_1(t)}$ is the position in meters of car one after time t in seconds and ${\displaystyle x_2(t)}$ is the position of car two after time t.

${\displaystyle x_1(t)=d+v_{1}t=200+22t,x_2(t)=v_{2}t=30t.}$

Notice that these formulas predict at t = 0 s the first car is 200m down the road and the second car is right beside us, as expected. We want to find the time at which ${\displaystyle x_1=x_2}$. Therefore, we set ${\displaystyle x_1=x_2}$ and solve for ${\displaystyle t}$, that is:

${\displaystyle 200+22t=30t,}$

${\displaystyle 8t=200,}$

${\displaystyle t=25\mathrm{s}\mathrm{e}\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}\mathrm{s}.}$

Alternatively, we could choose a frame of reference S′ situated in the first car. In this case, the first car is stationary and the second car is approaching from behind at a speed of v₂ − v₁ = 8 m/s. To catch up to the first car, it will take a time of ⁠d/v₂ − v₁⁠ = ⁠200/8⁠ s, that is, 25 seconds, as before. Note how much easier the problem becomes by choosing a suitable frame of reference. The third possible frame of reference would be attached to the second car. That example resembles the case just discussed, except the second car is stationary and the first car moves backward towards it at 8 m/s.

It would have been possible to choose a rotating, accelerating frame of reference, moving in a complicated manner, but this would have served to complicate the problem unnecessarily. One can convert measurements made in one coordinate system to another. For example, suppose that your watch is running five minutes fast compared to the local standard time. If you know that this is the case, when somebody asks you what time it is, you can deduct five minutes from the time displayed on your watch to obtain the correct time. The measurements that an observer makes about a system depend therefore on the observer's frame of reference (you might say that the bus arrived at 5 past three, when in fact it arrived at three).

Additional example

Figure 2: Simple-minded frame-of-reference example

For a simple example involving only the orientation of two observers, consider two people standing, facing each other on either side of a north-south street. See Figure 2. A car drives past them heading south. For the person facing east, the car was moving to the right. However, for the person facing west, the car was moving to the left. This discrepancy is because the two people used two different frames of reference from which to investigate this system.

For a more complex example involving observers in relative motion, consider Alfred, who is standing on the side of a road watching a car drive past him from left to right. In his frame of reference, Alfred defines the spot where he is standing as the origin, the road as the x-axis, and the direction in front of him as the positive y-axis. To him, the car moves along the x axis with some velocity v in the positive x-direction. Alfred's frame of reference is considered an inertial frame because he is not accelerating, ignoring effects such as Earth's rotation and gravity.

Now consider Betsy, the person driving the car. Betsy, in choosing her frame of reference, defines her location as the origin, the direction to her right as the positive x-axis, and the direction in front of her as the positive y-axis. In this frame of reference, it is Betsy who is stationary and the world around her that is moving – for instance, as she drives past Alfred, she observes him moving with velocity v in the negative y-direction. If she is driving north, then north is the positive y-direction; if she turns east, east becomes the positive y-direction.

Finally, as an example of non-inertial observers, assume Candace is accelerating her car. As she passes by him, Alfred measures her acceleration and finds it to be a in the negative x-direction. Assuming Candace's acceleration is constant, what acceleration does Betsy measure? If Betsy's velocity v is constant, she is in an inertial frame of reference, and she will find the acceleration to be the same as Alfred in her frame of reference, a in the negative y-direction. However, if she is accelerating at rate A in the negative y-direction (in other words, slowing down), she will find Candace's acceleration to be a′ = a − A in the negative y-direction—a smaller value than Alfred has measured. Similarly, if she is accelerating at rate A in the positive y-direction (speeding up), she will observe Candace's acceleration as a′ = a + A in the negative y-direction—a larger value than Alfred's measurement.

Non-inertial frames

Main articles: Fictitious force, Non-inertial frame, and Rotating frame of reference

Here the relation between inertial and non-inertial observational frames of reference is considered. The basic difference between these frames is the need in non-inertial frames for fictitious forces, as described below.

General relativity

Main articles: General relativity and Introduction to general relativity

See also: Equivalence principle and Eötvös experiment

General relativity is based upon the principle of equivalence:

There is no experiment observers can perform to distinguish whether an acceleration arises because of a gravitational force or because their reference frame is accelerating.

— Douglas C. Giancoli, Physics for Scientists and Engineers with Modern Physics, p. 155.

This idea was introduced in Einstein's 1907 article "Principle of Relativity and Gravitation" and later developed in 1911. Support for this principle is found in the Eötvös experiment, which determines whether the ratio of inertial to gravitational mass is the same for all bodies, regardless of size or composition. To date no difference has been found to a few parts in 10¹¹. For some discussion of the subtleties of the Eötvös experiment, such as the local mass distribution around the experimental site (including a quip about the mass of Eötvös himself), see Franklin.

Einstein's general theory modifies the distinction between nominally "inertial" and "non-inertial" effects by replacing special relativity's "flat" Minkowski Space with a metric that produces non-zero curvature. In general relativity, the principle of inertia is replaced with the principle of geodesic motion, whereby objects move in a way dictated by the curvature of spacetime. As a consequence of this curvature, it is not a given in general relativity that inertial objects moving at a particular rate with respect to each other will continue to do so. This phenomenon of geodesic deviation means that inertial frames of reference do not exist globally as they do in Newtonian mechanics and special relativity.

However, the general theory reduces to the special theory over sufficiently small regions of spacetime, where curvature effects become less important and the earlier inertial frame arguments can come back into play. Consequently, modern special relativity is now sometimes described as only a "local theory". "Local" can encompass, for example, the entire Milky Way galaxy: The astronomer Karl Schwarzschild observed the motion of pairs of stars orbiting each other. He found that the two orbits of the stars of such a system lie in a plane, and the perihelion of the orbits of the two stars remains pointing in the same direction with respect to the Solar System. Schwarzschild pointed out that that was invariably seen: the direction of the angular momentum of all observed double star systems remains fixed with respect to the direction of the angular momentum of the Solar System. These observations allowed him to conclude that inertial frames inside the galaxy do not rotate with respect to one another, and that the space of the Milky Way is approximately Galilean or Minkowskian.

Inertial frames and rotation

In an inertial frame, Newton's first law, the law of inertia, is satisfied: Any free motion has a constant magnitude and direction. Newton's second law for a particle takes the form:

${\displaystyle \mathbf{F}=m\mathbf{a},}$

with F the net force (a vector), m the mass of a particle and a the acceleration of the particle (also a vector) which would be measured by an observer at rest in the frame. The force F is the vector sum of all "real" forces on the particle, such as contact forces, electromagnetic, gravitational, and nuclear forces.

In contrast, Newton's second law in a rotating frame of reference (a non-inertial frame of reference), rotating at angular rate Ω about an axis, takes the form:

${\displaystyle {\mathbf{F}}^{\prime }=m\mathbf{a},}$

which looks the same as in an inertial frame, but now the force F′ is the resultant of not only F, but also additional terms (the paragraph following this equation presents the main points without detailed mathematics):

${\displaystyle {\mathbf{F}}^{\prime }=\mathbf{F}-2m\mathbf{\Omega }\times {\mathbf{v}}_B-m\mathbf{\Omega }\times (\mathbf{\Omega }\times {\mathbf{x}}_B)-m\frac{d\mathbf{\Omega }}{dt}\times {\mathbf{x}}_B,}$

where the angular rotation of the frame is expressed by the vector Ω pointing in the direction of the axis of rotation, and with magnitude equal to the angular rate of rotation Ω, symbol × denotes the vector cross product, vector x_B locates the body and vector v_B is the velocity of the body according to a rotating observer (different from the velocity seen by the inertial observer).

The extra terms in the force F′ are the "fictitious" forces for this frame, whose causes are external to the system in the frame. The first extra term is the Coriolis force, the second the centrifugal force, and the third the Euler force. These terms all have these properties: they vanish when Ω = 0; that is, they are zero for an inertial frame (which, of course, does not rotate); they take on a different magnitude and direction in every rotating frame, depending upon its particular value of Ω; they are ubiquitous in the rotating frame (affect every particle, regardless of circumstance); and they have no apparent source in identifiable physical sources, in particular, matter. Also, fictitious forces do not drop off with distance (unlike, for example, nuclear forces or electrical forces). For example, the centrifugal force that appears to emanate from the axis of rotation in a rotating frame increases with distance from the axis.

All observers agree on the real forces, F; only non-inertial observers need fictitious forces. The laws of physics in the inertial frame are simpler because unnecessary forces are not present.

In Newton's time the fixed stars were invoked as a reference frame, supposedly at rest relative to absolute space. In reference frames that were either at rest with respect to the fixed stars or in uniform translation relative to these stars, Newton's laws of motion were supposed to hold. In contrast, in frames accelerating with respect to the fixed stars, an important case being frames rotating relative to the fixed stars, the laws of motion did not hold in their simplest form, but had to be supplemented by the addition of fictitious forces, for example, the Coriolis force and the centrifugal force. Two experiments were devised by Newton to demonstrate how these forces could be discovered, thereby revealing to an observer that they were not in an inertial frame: the example of the tension in the cord linking two spheres rotating about their center of gravity, and the example of the curvature of the surface of water in a rotating bucket. In both cases, application of Newton's second law would not work for the rotating observer without invoking centrifugal and Coriolis forces to account for their observations (tension in the case of the spheres; parabolic water surface in the case of the rotating bucket).

As now known, the fixed stars are not fixed. Those that reside in the Milky Way turn with the galaxy, exhibiting proper motions. Those that are outside our galaxy (such as nebulae once mistaken to be stars) participate in their own motion as well, partly due to expansion of the universe, and partly due to peculiar velocities. For instance, the Andromeda Galaxy is on collision course with the Milky Way at a speed of 117 km/s. The concept of inertial frames of reference is no longer tied to either the fixed stars or to absolute space. Rather, the identification of an inertial frame is based on the simplicity of the laws of physics in the frame. The laws of nature take a simpler form in inertial frames of reference because in these frames one did not have to introduce inertial forces when writing down Newton's law of motion.

In practice, using a frame of reference based upon the fixed stars as though it were an inertial frame of reference introduces little discrepancy. For example, the centrifugal acceleration of the Earth because of its rotation about the Sun is about thirty million times greater than that of the Sun about the galactic center.

To illustrate further, consider the question: "Does the Universe rotate?" An answer might explain the shape of the Milky Way galaxy using the laws of physics, although other observations might be more definitive; that is, provide larger discrepancies or less measurement uncertainty, like the anisotropy of the microwave background radiation or Big Bang nucleosynthesis. The flatness of the Milky Way depends on its rate of rotation in an inertial frame of reference. If its apparent rate of rotation is attributed entirely to rotation in an inertial frame, a different "flatness" is predicted than if it is supposed that part of this rotation is actually due to rotation of the universe and should not be included in the rotation of the galaxy itself. Based upon the laws of physics, a model is set up in which one parameter is the rate of rotation of the Universe. If the laws of physics agree more accurately with observations in a model with rotation than without it, we are inclined to select the best-fit value for rotation, subject to all other pertinent experimental observations. If no value of the rotation parameter is successful and theory is not within observational error, a modification of physical law is considered, for example, dark matter is invoked to explain the galactic rotation curve. So far, observations show any rotation of the universe is very slow, no faster than once every 6×10¹³ years (10⁻¹³ rad/yr), and debate persists over whether there is any rotation. However, if rotation were found, interpretation of observations in a frame tied to the universe would have to be corrected for the fictitious forces inherent in such rotation in classical physics and special relativity, or interpreted as the curvature of spacetime and the motion of matter along the geodesics in general relativity.

When quantum effects are important, there are additional conceptual complications that arise in quantum reference frames.

Primed frames

An accelerated frame of reference is often delineated as being the "primed" frame, and all variables that are dependent on that frame are notated with primes, e.g. x′, y′, a′.

The vector from the origin of an inertial reference frame to the origin of an accelerated reference frame is commonly notated as R. Given a point of interest that exists in both frames, the vector from the inertial origin to the point is called r, and the vector from the accelerated origin to the point is called r′.

From the geometry of the situation

${\displaystyle \mathbf{r}=\mathbf{R}+{\mathbf{r}}^{\prime }.}$

Taking the first and second derivatives of this with respect to time

${\displaystyle \mathbf{v}=\mathbf{V}+{\mathbf{v}}^{\prime },}$

${\displaystyle \mathbf{a}=\mathbf{A}+{\mathbf{a}}^{\prime }.}$

where V and A are the velocity and acceleration of the accelerated system with respect to the inertial system and v and a are the velocity and acceleration of the point of interest with respect to the inertial frame.

These equations allow transformations between the two coordinate systems; for example, Newton's second law can be written as

${\displaystyle \mathbf{F}=m\mathbf{a}=m\mathbf{A}+m{\mathbf{a}}^{\prime }.}$

When there is accelerated motion due to a force being exerted there is manifestation of inertia. If an electric car designed to recharge its battery system when decelerating is switched to braking, the batteries are recharged, illustrating the physical strength of manifestation of inertia. However, the manifestation of inertia does not prevent acceleration (or deceleration), for manifestation of inertia occurs in response to change in velocity due to a force. Seen from the perspective of a rotating frame of reference the manifestation of inertia appears to exert a force (either in centrifugal direction, or in a direction orthogonal to an object's motion, the Coriolis effect).

A common sort of accelerated reference frame is a frame that is both rotating and translating (an example is a frame of reference attached to a CD which is playing while the player is carried).

This arrangement leads to the equation (see Fictitious force for a derivation):

${\displaystyle \mathbf{a}={\mathbf{a}}^{\prime }+\dot{\mathit{\omega }}\times {\mathbf{r}}^{\prime }+2\mathit{\omega }\times {\mathbf{v}}^{\prime }+\mathit{\omega }\times (\mathit{\omega }\times {\mathbf{r}}^{\prime })+{\mathbf{A}}_0,}$

or, to solve for the acceleration in the accelerated frame,

${\displaystyle {\mathbf{a}}^{\prime }=\mathbf{a}-\dot{\mathit{\omega }}\times {\mathbf{r}}^{\prime }-2\mathit{\omega }\times {\mathbf{v}}^{\prime }-\mathit{\omega }\times (\mathit{\omega }\times {\mathbf{r}}^{\prime })-{\mathbf{A}}_0.}$

Multiplying through by the mass m gives

${\displaystyle {\mathbf{F}}^{\prime }={\mathbf{F}}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{l}}+{\mathbf{F}}_{\mathrm{E}\mathrm{u}\mathrm{l}\mathrm{e}\mathrm{r}}^{\prime }+{\mathbf{F}}_{\mathrm{C}\mathrm{o}\mathrm{r}\mathrm{i}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{s}}^{\prime }+{\mathbf{F}}_{\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{p}\mathrm{e}\mathrm{t}\mathrm{a}\mathrm{l}}^{\prime }-m{\mathbf{A}}_0,}$

where

${\displaystyle {\mathbf{F}}_{\mathrm{E}\mathrm{u}\mathrm{l}\mathrm{e}\mathrm{r}}^{\prime }=-m\dot{\mathit{\omega }}\times {\mathbf{r}}^{\prime }}$ (Euler force),

${\displaystyle {\mathbf{F}}_{\mathrm{C}\mathrm{o}\mathrm{r}\mathrm{i}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{s}}^{\prime }=-2m\mathit{\omega }\times {\mathbf{v}}^{\prime }}$ (Coriolis force),

${\displaystyle {\mathbf{F}}_{\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{f}\mathrm{u}\mathrm{g}\mathrm{a}\mathrm{l}}^{\prime }=-m\mathit{\omega }\times (\mathit{\omega }\times {\mathbf{r}}^{\prime })=m({\omega }^2{\mathbf{r}}^{\prime }-(\mathit{\omega }\cdot {\mathbf{r}}^{\prime })\mathit{\omega })}$ (centrifugal force).

Separating non-inertial from inertial reference frames

Theory

Main article: Fictitious force

See also: Non-inertial frame, Rotating spheres, and Bucket argument

Figure 2: Two spheres tied with a string and rotating at an angular rate ω. Because of the rotation, the string tying the spheres together is under tension.

Figure 3: Exploded view of rotating spheres in an inertial frame of reference showing the centripetal forces on the spheres provided by the tension in the tying string.

Inertial and non-inertial reference frames can be distinguished by the absence or presence of fictitious forces.

The effect of this being in the noninertial frame is to require the observer to introduce a fictitious force into his calculations…

— Sidney Borowitz and Lawrence A Bornstein in A Contemporary View of Elementary Physics, p. 138

The presence of fictitious forces indicates the physical laws are not the simplest laws available, in terms of the special principle of relativity, a frame where fictitious forces are present is not an inertial frame:

The equations of motion in a non-inertial system differ from the equations in an inertial system by additional terms called inertial forces. This allows us to detect experimentally the non-inertial nature of a system.

— V. I. Arnol'd: Mathematical Methods of Classical Mechanics Second Edition, p. 129

Bodies in non-inertial reference frames are subject to so-called fictitious forces (pseudo-forces); that is, forces that result from the acceleration of the reference frame itself and not from any physical force acting on the body. Examples of fictitious forces are the centrifugal force and the Coriolis force in rotating reference frames.

To apply the Newtonian definition of an inertial frame, the understanding of separation between "fictitious" forces and "real" forces must be made clear.

For example, consider a stationary object in an inertial frame. Being at rest, no net force is applied. But in a frame rotating about a fixed axis, the object appears to move in a circle, and is subject to centripetal force. How can it be decided that the rotating frame is a non-inertial frame? There are two approaches to this resolution: one approach is to look for the origin of the fictitious forces (the Coriolis force and the centrifugal force). It will be found there are no sources for these forces, no associated force carriers, no originating bodies. A second approach is to look at a variety of frames of reference. For any inertial frame, the Coriolis force and the centrifugal force disappear, so application of the principle of special relativity would identify these frames where the forces disappear as sharing the same and the simplest physical laws, and hence rule that the rotating frame is not an inertial frame.

Newton examined this problem himself using rotating spheres, as shown in Figure 2 and Figure 3. He pointed out that if the spheres are not rotating, the tension in the tying string is measured as zero in every frame of reference. If the spheres only appear to rotate (that is, we are watching stationary spheres from a rotating frame), the zero tension in the string is accounted for by observing that the centripetal force is supplied by the centrifugal and Coriolis forces in combination, so no tension is needed. If the spheres really are rotating, the tension observed is exactly the centripetal force required by the circular motion. Thus, measurement of the tension in the string identifies the inertial frame: it is the one where the tension in the string provides exactly the centripetal force demanded by the motion as it is observed in that frame, and not a different value. That is, the inertial frame is the one where the fictitious forces vanish.

For linear acceleration, Newton expressed the idea of undetectability of straight-line accelerations held in common:

If bodies, any how moved among themselves, are urged in the direction of parallel lines by equal accelerative forces, they will continue to move among themselves, after the same manner as if they had been urged by no such forces.

— Isaac Newton: Principia Corollary VI, p. 89, in Andrew Motte translation

This principle generalizes the notion of an inertial frame. For example, an observer confined in a free-falling lift will assert that he himself is a valid inertial frame, even if he is accelerating under gravity, so long as he has no knowledge about anything outside the lift. So, strictly speaking, inertial frame is a relative concept. With this in mind, inertial frames can collectively be defined as a set of frames which are stationary or moving at constant velocity with respect to each other, so that a single inertial frame is defined as an element of this set.

For these ideas to apply, everything observed in the frame has to be subject to a base-line, common acceleration shared by the frame itself. That situation would apply, for example, to the elevator example, where all objects are subject to the same gravitational acceleration, and the elevator itself accelerates at the same rate.

Applications

Inertial navigation systems used a cluster of gyroscopes and accelerometers to determine accelerations relative to inertial space. After a gyroscope is spun up in a particular orientation in inertial space, the law of conservation of angular momentum requires that it retain that orientation as long as no external forces are applied to it. Three orthogonal gyroscopes establish an inertial reference frame, and the accelerators measure acceleration relative to that frame. The accelerations, along with a clock, can then be used to calculate the change in position. Thus, inertial navigation is a form of dead reckoning that requires no external input, and therefore cannot be jammed by any external or internal signal source.

A gyrocompass, employed for navigation of seagoing vessels, finds the geometric north. It does so, not by sensing the Earth's magnetic field, but by using inertial space as its reference. The outer casing of the gyrocompass device is held in such a way that it remains aligned with the local plumb line. When the gyroscope wheel inside the gyrocompass device is spun up, the way the gyroscope wheel is suspended causes the gyroscope wheel to gradually align its spinning axis with the Earth's axis. Alignment with the Earth's axis is the only direction for which the gyroscope's spinning axis can be stationary with respect to the Earth and not be required to change direction with respect to inertial space. After being spun up, a gyrocompass can reach the direction of alignment with the Earth's axis in as little as a quarter of an hour.