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Victoria (TV series)
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For the telenovela, see Victoria (telenovela).
Victoria
Genre
Drama
Created by
Daisy Goodwin
Written by
Daisy Goodwin
Guy Andrews
Ottilie Wilford
Starring
Jenna Coleman
Tom Hughes
Peter Bowles
Catherine Flemming
Daniela Holtz
Nell Hudson
Ferdinand Kingsley
Tommy Knight
Nigel Lindsay
Eve Myles
David Oakes
Paul Rhys
Adrian Schiller
Peter Firth
Alex Jennings
Rufus Sewell
Bebe Cave
Margaret Clunie
Tilly Steele
Leo Suter
Jordan Waller
Anna Wilson-Jones
Bruno Wolkowitch
Martin Compston
Denis Lawson
Diana Rigg
Composer(s)
Ruth Barrett
Martin Phipps
Country of origin
United Kingdom
Original language(s)
English
No. of series
2
No. of episodes
15 (list of episodes)
Production
Executive producer(s)
Daisy Goodwin
Dan McCulloch
Damien Timmer
Producer(s)
Rebecca Eaton
Paul Frift
Location(s)
United Kingdom
Running time
46–69 minutes
Production company(s)
Mammoth Screen
Distributor
ITV Studios
Release
Original network
ITV
Picture format
16:9 1080i
Audio format
Stereo
Original release
28 August 2016 (2016-08-28) – present (present)
External links
Website
www.itv.com/hub/victoria/
Victoria is a British drama television series created by Daisy Goodwin and stars Jenna Coleman as Queen Victoria. The series was announced in September 2015.
Victoria premiered on 28 August 2016 on ITV. The first series consisted of eight episodes, and concluded on 9 October 2016. In September 2016, ITV renewed Victoria for an eight-episode second series, which premiered on 27 August 2017, and will be followed by a 2017 Christmas special.
Contents
[hide]
1 Premise
2 Cast
2.1 Main
2.2 Recurring
3 Episodes
3.1 Series 1 (2016)
3.2 Series 2 (2017)
4 Music
5 Production
5.1 Filming
5.2 Development
6 Broadcast
7 Reception
8 References
9 External links
Premise[edit]
The first series depicts the first few years of the reign of Queen Victoria, portrayed by Jenna Coleman, from her accession to the throne at the age of eighteen, to her intense friendship and infatuation with Lord Melbourne, to her courtship and early marriage to Prince Albert (Tom Hughes) and finally to the birth of their first child, Victoria.
The second series follows Victoria's struggles on managing her role as Queen with that of her duties to her husband and children.
Cast[edit]
Main[edit]
Jenna Coleman as Queen Victoria[1]
Tom Hughes as Prince Albert[2]
Peter Bowles as the Duke of Wellington
Catherine Flemming as the Duchess of Kent[1]
Daniela Holtz as Baroness Lehzen[2]
Nell Hudson as Miss Skerrett[3]
Ferdinand Kingsley as Charles Elmé Francatelli[4]
Tommy Knight as Archibald Brodie[5]
Nigel Lindsay as Sir Robert Peel[2]
Eve Myles as Mrs. Jenkins[6] (series 1)
David Oakes as Prince Ernest
Paul Rhys as Sir John Conroy[1]
Adrian Schiller as Mr. Penge[3]
Peter Firth as the Duke of Cumberland and Teviotdale[1]
Alex Jennings as King Leopold
Rufus Sewell as Lord Melbourne[1] (series 1–2)
Bebe Cave as Wilhelmina Coke (series 2–)
Margaret Clunie as the Duchess of Sutherland (recurring series 1, starring series 2–)
Tilly Steele as Miss Cleary (series 2–)
Leo Suter as Edward Drummond (series 2–)
Jordan Waller as Lord Alfred Paget (recurring series 1, starring series 2–)
Anna Wilson-Jones as Lady Portman (recurring series 1, starring series 2–)
Bruno Wolkowitch as King Louis Philippe (series 2)
Martin Compston as Dr. Traill (series 2)
Denis Lawson as the Duke of Atholl (series 2)
Diana Rigg as Duchess of Buccleuch (series 2–)
Recurring[edit]
Tom Price as the Duke of Sutherland
Robin McCallum as Lord Portman
Richard Dixon as the Lord Chancellor
Julian Finnigan as Lord Hastings
Guy Oliver-Watts as Sir James Hayter
Michael Parkhouse as Lord Melbourne's butler
Alice Orr-Ewing as Lady Flora Hastings
Nichola McAuliffe as the Duchess of Cumberland, the German wife of the Duke of Cumberland[2]
Simon Paisley Day as Lord Chamberlain
Basil Eidenbenz as Lohlein, Prince Albert's valet
Harry McEntire as Edward Oxford
Andrew Scarborough as Captain Childers
Aoife Kennan as Abigail Owen
Samantha Colley as Eliza Skerrett
Robin Soans as Sir James Clark
Bruce Mackinnon as Hannam
James Wilby as Sir Piers Gifford
Annabel Mullion as Lady Beatrice Gifford
Daisy Brown as Soprano
Daniel Donskoy as The Grand Duke
Nicholas Agnew as Prince George
Andrew Bicknell as the Duke of Coburg
Carolin Stoltz as Gretchen
Ben Abell as Rowland Hill
Gertrude Thoma as Hilde
Robert MacPherson as Anson
David Bamber as the Duke of Sussex
Terence Beesley as Buxton
Daisy Goodwin as the Duchess of Inverness
Cornell S. John as Jonas Barrett
Clare Wille as Lady Peel
Peter Ivatts as the Archbishop of Canterbury, William Howley.
Peter Forbes as Craddock
Alexander Owen as Dr Brydon
Phil Rowson as John Bright, MP
Episodes[edit]
Series
Episodes
Originally aired
First aired
Last aired
1 8 28 August 2016 (2016-08-28) 9 October 2016 (2016-10-09)
2 8 27 August 2017 (2017-08-27) 15 October 2017 (2017-10-15)[7]
Series 1 (2016)[edit]
No.
overall
No. in
series
Title
Directed by
Written by
Original air date
UK viewers
(millions) [8]
1
1 "Doll 123" Tom Vaughan Daisy Goodwin 28 August 2016 (2016-08-28) 8.00
Victoria ascends to the throne after the death of her paternal uncle, King William. She asserts her independence by standing up to her mother and Sir John Conroy, both had reared her under the strict Kensington System. Victoria creates a scandal involving her mother's lady-in-waiting, Lady Flora Hastings, by spreading the rumor that Lady Flora is pregnant with Sir John's child. Lady Flora is forced to undergo a medical examination, which reveals she is a virgin, but with a growing abdominal tumour that will prove fatal. The public is outraged when they learn of the humiliation the dying Lady Flora suffered at Victoria's hands. Victoria despairs over what she has done and feels her reign is tarnished, but is encouraged by the prime minister, Lord Melbourne, to stand tall and face the public.
2
2 "Ladies in Waiting" Tom Vaughan Daisy Goodwin 29 August 2016 (2016-08-29) 7.37
The Whig Prime Minister Lord Melbourne announces he must resign, saying he lacks the required mandate in parliament. Victoria asks the Duke of Wellington to form a new government but he declines, saying he is too old to be prime minister again. He suggests she instead invite Tory leader Sir Robert Peel, which she does reluctantly. However, Peel agrees only on the condition that she dismiss some of her Ladies of the Bedchamber, as four are married to Whig ministers, and replace them with the wives of Tories. The queen refuses, as she considers the ladies to be her intimate friends and not political pawns. This leads to the Bedchamber crisis. Melbourne insists he cannot undermine the British constitution by governing solely at the monarch's whim, but he eventually gives in and stays on as Prime Minister. Meanwhile, Victoria's mother, Sir John and her paternal uncle the Duke of Cumberland scheme to install a regency to limit Victoria's power by making people believe she had inherited the madness of her grandfather, King George III.
3
3 "Brocket Hall" Tom Vaughan Daisy Goodwin 4 September 2016 (2016-09-04) 7.75
Sir John and Victoria's mother want her to marry, believing she is too independent and needs a man to control her. Victoria's maternal uncle, King Leopold of Belgium, arrives to pressure her to marry his nephew and her cousin Prince Albert of Saxe-Coburg and Gotha. Victoria has no interest in Albert. Leopold tries to get Lord Melbourne to support the scheme; Melbourne believes there is no hurry to push her into a marriage if it is with the wrong person. Victoria, believing she is in love with Melbourne, pursues him, but he rebuffs her. Tired of Sir John, Victoria offers him a baronetcy and pension for him to leave court, devastating her mother. Following the Newport Rising, Victoria takes pity on the leaders and commutes their death sentences to exile in Australia.
4
4 "The Clockwork Prince" Sandra Goldbacher Daisy Goodwin 11 September 2016 (2016-09-11) 7.62
Albert arrives to visit Victoria. They are mutually aloof, and resentful for being pushed on each other. Some at court feel Albert, as a younger son and prince of a minor duchy, is beneath her. As they spend time together, an attraction slowly grows. Albert, however, challenges Victoria on Lord Melbourne's influence over her and his sheltering her from reality, such as the poverty of the masses. Following an argument, Albert prepares to leave, believing he has lost her favour, while Victoria herself has come to terms with her love for Albert but is hesitant to propose because she is no longer sure if he will say yes. After confiding her worries to Leopold and being given some encouragement, Victoria proposes and Albert eagerly accepts.
5
5 "An Ordinary Woman" Sandra Goldbacher Daisy Goodwin 18 September 2016 (2016-09-18) 7.65
When Victoria announces her engagement, the Tory parliament reacts with hostility at the idea of a German consort, especially one who would be given a British title and hefty annual allowance. Additionally, the Duke of Wellington calls attention to the certainty that Albert is not a Catholic, which would preclude Victoria from marrying him without abdicating her crown. Though unable to give him a royal title such as Duke or King, Victoria makes Albert a Knight of the Garter and gives him a small allowance. Albert is offended, feeling he will be without position or independence, while Victoria worries that the allowance would enable him to keep a mistress, as did her Uncle Leopold. They reassure each other and, on 10 February 1840, they are married.
6
6 "The Queen's Husband" Olly Blackburn Daisy Goodwin 25 September 2016 (2016-09-25) 7.65
Victoria and Albert are happily married. However, Victoria fears she will die in childbirth like her cousin and aunt Princess Charlotte, whose death led to Victoria inheriting the crown. Her lady's maid gives her useless advice to avoid pregnancy. Victoria curries favor with her paternal uncle the Duke of Sussex, who is unable to present his wife at court because their morganatic marriage was in violation of the Royal Marriages Act 1772. Although his wife is the daughter of an earl, she is not of royal birth. Victoria uses her discretion to make her the Duchess of Inverness and welcomes her to court. An abolitionist group asks Victoria to open its London meeting, held in June 1840. She says she will publicly support the cause, but is unable to open it because of her position. Prince Albert instead volunteers to give an address against slavery, a subject he feels strongly about. His speech is considered a great success for him.
7
7 "The Engine of Change" Olly Blackburn Guy Andrews 2 October 2016 (2016-10-02) 7.31
Victoria, pregnant with her first child, is desperately afraid of dying in childbirth. Her mother insists she refrain from affairs of state and rest full-time. The ministers ask Victoria to choose a regent in the case she dies in childbirth but her baby survives. She insists on Albert as regent, which enrages the Tories. However, Sir Robert Peel believes Albert could be influenced and supports him as regent. Albert takes an interest in the building of the railway to improve Great Britain, a project Victoria discourages him from pursuing. Sir Robert takes Albert to ride on his locomotive, which infuriates her. Albert, however, insists that he have some measure of influence in his new country, while she worries that his doing so could undermine her own authority as queen. Nevertheless Victoria bravely ventures out to take a ride on the locomotive herself, which she finds exhilarating.
8
8 "Young England" Olly Blackburn Daisy Goodwin 9 October 2016 (2016-10-09) 7.74
The pregnant Victoria insists on her daily carriage rides to greet her subjects. The Duke of Cumberland, now the King of Hanover, insists that the British people are unaccepting of the German Albert as regent should death result from childbirth. Cumberland hints of assassination threats. On 10 June 1840, Edward Oxford attempts to assassinate Victoria during a carriage ride. Cumberland, who is first in line to the throne if Victoria dies, is blamed but denies any involvement. Oxford's guns were unloaded at the time. He is declared not guilty by reason of insanity and sent to a lunatic asylum. On 21 November 1840, Victoria gives birth without complications to a healthy baby girl, also named Victoria.
Series 2 (2017)[edit]
No.
overall
No. in
series
Title
Directed by
Written by
Original air date
UK viewers
(millions) [8]
9
1 "A Soldier's Daughter" Lisa James Larsson Daisy Goodwin 27 August 2017 (2017-08-27) 6.17
10
2 "The Green-Eyed Monster" Lisa James Larsson Daisy Goodwin 3 September 2017 (2017-09-03) 6.62
11
3 "Warp and Weft" Geoffrey Sax Daisy Goodwin 10 September 2017 (2017-09-10) 6.56
12
4 "The Sins of the Father" Geoffrey Sax Ottilie Wilford 17 September 2017 (2017-09-17) 6.77
13
5 "Entente Cordiale" Jim Loach Daisy Goodwin 24 September 2017 (2017-09-24) 6.15
14
6 "Faith, Hope & Charity" Jim Loach Daisy Goodwin 1 October 2017 (2017-10-01) 6.05
15
7 "The King Over the Water" Daniel O'Hara Ottilie Wilford 8 October 2017 (2017-10-08) TBD
16
8 "The Luxury of Conscience"[7] TBA Daisy Goodwin 15 October 2017 (2017-10-15) TBD
Music[edit]
The theme song is by Martin Phipps, sung by the Mediæval Bæbes. Phipps also wrote and conducted incidental music for the early episodes. For later episodes the conducting role was undertaken by Ruth Barrett.[9] An official soundtrack for the first series was released on 12 January 2017.[10][11]
[show]Track listing
No.
Title
Performers
Length
1. "Victoria – The Suite" Benji Merrison, Mediæval Bæbes 3:27
2. "The King is Dead" 2:49
3. "Coronation" Mediæval Babes 3:15
4. "Lord M" 2:44
5. "Locomotives" 1:52
6. "Mirrors" 1:32
7. "The Wedding" Mediæval Babes 3:43
8. "The Royal Birth" 2:14
9. "Privy Council" 2:50
10. "A Royal Affair" 2:05
11. "Victoria Titles" Mediæval Babes 1:35
Total length: 28:03
Production[edit]
The series was announced in September 2015, following Coleman's decision to leave Doctor Who to join the cast as Queen Victoria.[12][13]
Filming[edit]
Much of Victoria was filmed in Yorkshire. The interiors of Castle Howard double as Kensington Palace, Harewood House stands in for Buckingham Palace, with Bramham Park and Wentworth Woodhouse also being in use for both royal residences. Carlton Towers is in use as Windsor Castle, while Beverley Minster replaces Westminster Abbey. Other locations include Raby Castle, Wentworth Woodhouse, Allerton Castle, Newby Hall and Whitby pier. Church Fenton Studios, a converted aircraft hangar at Leeds East Airport near Selby, was in use to recreate some interiors of the Buckingham Palace.[14][15]
Development[edit]
Daisy Goodwin said in October 2016 that a Christmas special episode for the 2016 season had been proposed and was rejected by ITV; one was subsequently commissioned for 2017 after the rising ratings popularity for Victoria.[16]
In September 2016, ITV renewed Victoria for a second series, followed by a Christmas special.[17][18] Beyond the renewal, ITV administration reportedly expects Victoria to run for a total of six series, although whether Jenna Coleman will remain as the lead actress as the series moves into depicting Victoria's later years is uncertain.[19]
Broadcast[edit]
The series premiered on 28 August 2016 on ITV in the UK, and on 15 January 2017 on PBS in the United States as part of Masterpiece.[20] The series premiered on 4 April 2017 in Canada on Vision TV.[21]
The eight-episode second series premiered on ITV on 27 August 2017 in the UK, and will premiere on 14 January 2018 on PBS in the United States.[22][23]
Reception[edit]
The critics' reviews, of the first series, have been positive. On Rotten Tomatoes the first season holds a rating of 81%, based on 37 reviews, with an average rating of 6,83/10. The site's consensus reads, "Strong performances by Jenna Coleman and Rufus Sewell hint at Downton-esque potential for Victoria, but the narrative falls just shy of that soapy mark".[24] On Metacritic season one has a score of 67 out of 100, based on 22 critics, indicating "generally favourable reviews".[25]
Year
Award
Category
Recipient
Result
Ref
2017 2017 British Academy Television Craft Awards Best Make Up and Hair Design Nic Collins Nominated [26]
69th Primetime Creative Arts Emmy Awards Outstanding Music Composition for a Series (Original Dramatic Score) Martin Phipps, Ruth Barrett, Natalie Holt for "Doll 123" Pending [27]
Outstanding Main Title Theme Music Martin Phipps Pending [27]
References[edit]
^ Jump up to: a b c d e "Rufus Sewell to join Jenna Coleman in major new ITV drama Victoria". ITV Press Centre. ITV. 21 October 2015.
^ Jump up to: a b c d "Tom Hughes to star as Prince Albert in major new ITV drama Victoria". ITV Press Centre. ITV. 14 December 2015.
^ Jump up to: a b Tartaglione, Nancy (21 October 2015). "Rufus Sewell Joins ITV's 'Victoria' As Period Drama Adds More Cast". Deadline Hollywood. Penske Business Media, LLC. Retrieved 29 August 2016.
Jump up ^ "It was impossible to turn down Victoria, says Jenna Coleman". The Indian Express. PTI. 14 August 2016. Retrieved 29 August 2016.
Jump up ^ Tommy Knight, Facebook
Jump up ^ Leane, Rob (22 October 2015). "Victoria: first pic of Jenna Coleman, Eve Myles joins cast". Den of Geek. Dennis Publishing Limited.
^ Jump up to: a b "Victoria Episode 8". ITV. Retrieved 6 October 2017.
^ Jump up to: a b "Weekly Viewing Summary (see relevant week)". BARB. The viewer figures take into account live viewing in SD and HD, +1 figures, recording viewers and ITV Hub viewers within 28 days after the broadcast.
Jump up ^ Dowell, Ben (25 September 2016). "What is the Victoria title music? Who wrote the theme? Who sings it? Was it played at her wedding to Albert?". Radio Times. Retrieved 25 September 2016.
Jump up ^ "Well we have rather excellent news... It's here... Head over to Spotify 👑 🎶 #Victoria #Soundtrack". Facebook. 13 January 2017. Retrieved 18 January 2017.
Jump up ^ "Victoria (Original Soundtrack)". Spotify. Retrieved 18 January 2017.
Jump up ^ "Jenna Coleman is Leaving Doctor Who". BBC. 18 September 2015. Retrieved 18 September 2015.
Jump up ^ Barraclough, Leo (18 September 2015). "Jenna Coleman Leaving 'Doctor Who' to Play Queen Victoria". Variety. Retrieved 18 June 2016.
Jump up ^ "Where was Victoria filmed?". Radio Times. 25 September 2016.
Jump up ^ Penfold, Phil (27 August 2016). "Filmed in Yorkshire, could Victoria fill the gap left by Downton?". The Yorkshire Post.
Jump up ^ Frost, Caroline (14 October 2016). "'Victoria' Drama Christmas Offer Turned Down By ITV, Reveals Creator Daisy Godwin". Huffington Post.
Jump up ^ Dowell, Ben (23 September 2016). "Hallelujah! Victoria to return for a second series". RadioTimes. Retrieved 23 September 2016.
Jump up ^ Lazarus, Susanna (6 October 2016). "ITV's Victoria gets a 2017 Christmas special, reveals screenwriter Daisy Goodwin". RadioTimes. Retrieved 9 October 2016.
Jump up ^ Frost, Caroline (11 October 2016). "'Victoria': ITV Bosses 'Considering Six Series in Total, But Swapping Jenna Coleman Halfway Through'". Huffington Post.
Jump up ^ Martin, William (12 August 2016). "US start date confirmed for Jenna Coleman's 'Victoria' series". CultBox. Retrieved 7 September 2016.
Jump up ^ "Episodes - Vision TV Channel Canada". Vision TV Canada. Retrieved 13 May 2017.
Jump up ^ Doran, Sarah (17 August 2017). "JK Rowling and Jenna Coleman set for Bank Holiday Sunday clash". Radio Times. Retrieved 18 August 2017.
Jump up ^ Petski, Denise (16 August 2017). "'Victoria' Gets Season 2 Premiere Date On PBS". Deadline. Retrieved 17 August 2017.
Jump up ^ "Victoria: Season 1 (2016)". www.rottentomatoes.com. Retrieved 13 May 2017.
Jump up ^ "Victoria". Metacritic.
Jump up ^ "Television Craft Awards Winners 2017". British Academy of Film and Television Arts (BAFTA). Retrieved 19 June 2017.
^ Jump up to: a b "69th Emmy® Awards Nominations" (PDF). Academy of Television Arts & Sciences. July 13, 2017. Retrieved July 13, 2017.
External links[edit]
Victoria on IMDb
Victoria at TV.com
Victoria at epguides.com
Retrieved from "https://en.wikipedia.org/w/index.php?title=Victoria_(TV_series)&oldid=804660157"
Categories:
2016 British television programme debuts
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Cultural depictions of Queen Victoria on television
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Television shows set in the United Kingdom
Hidden categories:
Use dmy dates from July 2017
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G μ ν + Λ g μ ν = 8 π G c 4 T μ ν {\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }={8\pi G \over c^{4}}T_{\mu \nu }}
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The Einstein field equations (EFE; also known as Einstein's equations) comprise the set of 10 equations in Albert Einstein's general theory of relativity that describe the fundamental interaction of gravitation as a result of spacetime being curved by mass and energy.[1] First published by Einstein in 1915[2] as a tensor equation, the EFE relate local spacetime curvature (expressed by the Einstein tensor) with the local energy and momentum within that spacetime (expressed by the stress–energy tensor).[3]
Similar to the way that electromagnetic fields are determined using charges and currents via Maxwell's equations, the EFE are used to determine the spacetime geometry resulting from the presence of mass–energy and linear momentum, that is, they determine the metric tensor of spacetime for a given arrangement of stress–energy in the spacetime. The relationship between the metric tensor and the Einstein tensor allows the EFE to be written as a set of non-linear 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 obeying local energy–momentum conservation, the EFE reduce to Newton's law of gravitation where the gravitational field is weak and velocities are much less than the speed of light.[4]
Exact solutions for the EFE can only be found under simplifying assumptions such as symmetry. Special classes of exact solutions are most often studied as they model many gravitational phenomena, such as rotating black holes and the expanding universe. Further simplification is achieved in approximating the actual spacetime as flat spacetime with a small deviation, leading to the linearized EFE. These equations are used to study phenomena such as gravitational waves.
Contents
[hide]
1 Mathematical form
1.1 Sign convention
1.2 Equivalent formulations
2 The cosmological constant
3 Features
3.1 Conservation of energy and momentum
3.2 Nonlinearity
3.3 The correspondence principle
4 Vacuum field equations
5 Einstein–Maxwell equations
6 Solutions
7 The linearized EFE
8 Polynomial form
9 See also
10 Notes
11 References
12 External links
Mathematical form[edit]
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Introduction to general relativity
Mathematics of general relativity
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Introduction to gravitation
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The Einstein field equations (EFE) may be written in the form:[5][1]
R μ ν − 1 2 R g μ ν + Λ g μ ν = 8 π G c 4 T μ ν {\displaystyle R_{\mu \nu }-{\tfrac {1}{2}}R\,g_{\mu \nu }+\Lambda g_{\mu \nu }={\frac {8\pi G}{c^{4}}}T_{\mu \nu }}
EFE on a wall in Leiden
where Rμν is the Ricci curvature tensor, R is the scalar curvature, gμν is the metric tensor, Λ is the cosmological constant, G is Newton's gravitational constant, c is the speed of light in vacuum, and Tμν is the stress–energy tensor.
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.[6] The equations in contexts outside of general relativity are still referred to as the Einstein field equations. The vacuum field equations (obtained when T is identically zero) define Einstein manifolds.
Despite the simple appearance of the equations they are actually quite complicated. 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 gμν, as both the Ricci tensor and scalar curvature depend on the metric in a complicated nonlinear manner. In fact, when fully written out, the EFE are a system of ten coupled, nonlinear, hyperbolic-elliptic partial differential equations.[citation needed]
One can write the EFE in a more compact form by defining the Einstein tensor
G μ ν = R μ ν − 1 2 R g μ ν , {\displaystyle G_{\mu \nu }=R_{\mu \nu }-{\tfrac {1}{2}}Rg_{\mu \nu },}
which is a symmetric second-rank tensor that is a function of the metric. The EFE can then be written as
G μ ν + Λ g μ ν = 8 π G c 4 T μ ν . {\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }={\frac {8\pi G}{c^{4}}}T_{\mu \nu }.}
Using geometrized units where G = c = 1, this can be rewritten as
G μ ν + Λ g μ ν = 8 π T μ ν . {\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }=8\pi T_{\mu \nu }\,.}
The expression on the left represents the curvature of spacetime as determined by the metric; the expression on the right represents the matter/energy content of spacetime. The EFE can then be interpreted as a set of equations dictating how matter/energy determines the curvature of spacetime.
These equations, together with the geodesic equation,[7] which dictates how freely-falling matter moves through space-time, form the core of the mathematical formulation of general relativity.
Sign convention[edit]
The above form of the EFE is the standard established by Misner, Thorne, and Wheeler.[8] The authors analyzed all conventions that exist and classified according to the following three signs (S1, S2, S3):
g μ ν = [ S 1 ] × diag  ( − 1 , + 1 , + 1 , + 1 ) R μ α β γ = [ S 2 ] × ( Γ α γ , β μ − Γ α β , γ μ + Γ σ β μ Γ γ α σ − Γ σ γ μ Γ β α σ ) G μ ν = [ S 3 ] × 8 π G c 4 T μ ν {\displaystyle {\begin{aligned}g_{\mu \nu }&=[S1]\times \operatorname {diag} (-1,+1,+1,+1)\\[6pt]{R^{\mu }}_{\alpha \beta \gamma }&=[S2]\times \left(\Gamma _{\alpha \gamma ,\beta }^{\mu }-\Gamma _{\alpha \beta ,\gamma }^{\mu }+\Gamma _{\sigma \beta }^{\mu }\Gamma _{\gamma \alpha }^{\sigma }-\Gamma _{\sigma \gamma }^{\mu }\Gamma _{\beta \alpha }^{\sigma }\right)\\[6pt]G_{\mu \nu }&=[S3]\times {\frac {8\pi G}{c^{4}}}T_{\mu \nu }\end{aligned}}}
The third sign above is related to the choice of convention for the Ricci tensor:
R μ ν = [ S 2 ] × [ S 3 ] × R α μ α ν {\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)[9] and Peacock (1994)[10] are (+ − −), Peebles (1980)[citation needed] and Efstathiou (1990)[citation needed] are (− + +), Rindler (1977)[citation needed], Atwater (1974)[citation needed], Collins Martin & Squires (1989)[citation needed] 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
R μ ν − 1 2 R g μ ν − Λ g μ ν = − 8 π G c 4 T μ ν . {\displaystyle R_{\mu \nu }-{\tfrac {1}{2}}Rg_{\mu \nu }-\Lambda g_{\mu \nu }=-{\frac {8\pi G}{c^{4}}}T_{\mu \nu }.}
The sign of the (very small) 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[edit]
Taking the trace with respect to the metric of both sides of the EFE one gets
R − D 2 R + D Λ = 8 π G c 4 T {\displaystyle R-{\frac {D}{2}}R+D\Lambda ={\frac {8\pi G}{c^{4}}}T\,}
where D is the spacetime dimension. This expression can be rewritten as
− R + D Λ D 2 − 1 = 8 π G c 4 T D 2 − 1 . {\displaystyle -R+{\frac {D\Lambda }{{\frac {D}{2}}-1}}={\frac {8\pi G}{c^{4}}}{\frac {T}{{\frac {D}{2}}-1}}\,.}
If one adds −1/2gμν times this to the EFE, one gets the following equivalent "trace-reversed" form
R μ ν − Λ g μ ν D 2 − 1 = 8 π G c 4 ( T μ ν − 1 D − 2 T g μ ν ) . {\displaystyle R_{\mu \nu }-{\frac {\Lambda g_{\mu \nu }}{{\frac {D}{2}}-1}}={\frac {8\pi G}{c^{4}}}\left(T_{\mu \nu }-{\frac {1}{D-2}}Tg_{\mu \nu }\right).\,}
For example, in D = 4 dimensions this reduces to
R μ ν − Λ g μ ν = 8 π G c 4 ( T μ ν − 1 2 T g μ ν ) . {\displaystyle R_{\mu \nu }-\Lambda g_{\mu \nu }={\frac {8\pi G}{c^{4}}}\left(T_{\mu \nu }-{\tfrac {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 gμν in the expression on the right with the Minkowski metric without significant loss of accuracy).
The cosmological constant[edit]
Main article: Cosmological constant
Einstein modified his original field equations to include a cosmological constant term Λ proportional to the metric
R μ ν − 1 2 R g μ ν + Λ g μ ν = 8 π G c 4 T μ ν . {\displaystyle R_{\mu \nu }-{\tfrac {1}{2}}R\,g_{\mu \nu }+\Lambda g_{\mu \nu }={\frac {8\pi G}{c^{4}}}T_{\mu \nu }\,.}
Since Λ is constant, the energy conservation law is unaffected.
The cosmological constant term was originally introduced by Einstein to allow for a universe that is not expanding or contracting. This effort was unsuccessful because:
the universe described by this theory was unstable, and
observations by Edwin Hubble confirmed that our universe is expanding.
So, Einstein abandoned Λ, calling it the "biggest blunder [he] ever made".[11]
Despite Einstein's motivation for introducing the cosmological constant term, there is nothing inconsistent with the presence of such a term in the equations. For many years the cosmological constant was almost universally considered to be 0. However, recent improved astronomical techniques have found that a positive value of Λ is needed to explain the accelerating universe.[12][13]
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, written as part of the stress–energy tensor:
T μ ν ( v a c ) = − Λ c 4 8 π G g μ ν . {\displaystyle T_{\mu \nu }^{\mathrm {(vac)} }=-{\frac {\Lambda c^{4}}{8\pi G}}g_{\mu \nu }\,.}
The resulting vacuum energy density is constant and given by
ρ v a c = Λ c 2 8 π G {\displaystyle \rho _{\mathrm {vac} }={\frac {\Lambda c^{2}}{8\pi G}}}
The existence of a cosmological constant is thus equivalent to the existence of a non-zero vacuum energy. Thus, the terms "cosmological constant" and "vacuum energy" are now used interchangeably in general relativity.
Features[edit]
Conservation of energy and momentum[edit]
General relativity is consistent with the local conservation of energy and momentum expressed as
∇ β T α β = T α β ; β = 0 {\displaystyle \nabla _{\beta }T^{\alpha \beta }={T^{\alpha \beta }}_{;\beta }=0} .
[show]Derivation of local energy-momentum conservation
Contracting the differential Bianchi identity
R α β [ γ δ ; ε ] = 0 {\displaystyle R_{\alpha \beta [\gamma \delta ;\varepsilon ]}=0}
with gαβ gives, using the fact that the metric tensor is covariantly constant, i.e. gαβ;γ = 0,
R γ β γ δ ; ε + R γ β ε γ ; δ + R γ β δ ε ; γ = 0 {\displaystyle {R^{\gamma }}_{\beta \gamma \delta ;\varepsilon }+{R^{\gamma }}_{\beta \varepsilon \gamma ;\delta }+{R^{\gamma }}_{\beta \delta \varepsilon ;\gamma }=\,0}
The antisymmetry of the Riemann tensor allows the second term in the above expression to be rewritten:
R γ β γ δ ; ε − R γ β γ ε ; δ + R γ β δ ε ; γ = 0 {\displaystyle {R^{\gamma }}_{\beta \gamma \delta ;\varepsilon }-{R^{\gamma }}_{\beta \gamma \varepsilon ;\delta }+{R^{\gamma }}_{\beta \delta \varepsilon ;\gamma }=0}
which is equivalent to
R β δ ; ε − R β ε ; δ + R γ β δ ε ; γ = 0 {\displaystyle R_{\beta \delta ;\varepsilon }-R_{\beta \varepsilon ;\delta }+{R^{\gamma }}_{\beta \delta \varepsilon ;\gamma }=0}
using the definition of the Ricci tensor.
Next, contract again with the metric
g β δ ( R β δ ; ε − R β ε ; δ + R γ β δ ε ; γ ) = 0 {\displaystyle g^{\beta \delta }\left(R_{\beta \delta ;\varepsilon }-R_{\beta \varepsilon ;\delta }+{R^{\gamma }}_{\beta \delta \varepsilon ;\gamma }\right)=0}
to get
R δ δ ; ε − R δ ε ; δ + R γ δ δ ε ; γ = 0 {\displaystyle {R^{\delta }}_{\delta ;\varepsilon }-{R^{\delta }}_{\varepsilon ;\delta }+{R^{\gamma \delta }}_{\delta \varepsilon ;\gamma }=0}
The definitions of the Ricci curvature tensor and the scalar curvature then show that
R ; ε − 2 R γ ε ; γ = 0 {\displaystyle R_{;\varepsilon }-2{R^{\gamma }}_{\varepsilon ;\gamma }=0}
which can be rewritten as
( R γ ε − 1 2 g γ ε R ) ; γ = 0 {\displaystyle \left({R^{\gamma }}_{\varepsilon }-{\tfrac {1}{2}}{g^{\gamma }}_{\varepsilon }R\right)_{;\gamma }=0}
A final contraction with gεδ gives
( R γ δ − 1 2 g γ δ R ) ; γ = 0 {\displaystyle \left(R^{\gamma \delta }-{\tfrac {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,
G α β ; β = 0 {\displaystyle {G^{\alpha \beta }}_{;\beta }=0}
Using the EFE, this immediately gives,
∇ β T α β = T α β ; β = 0 {\displaystyle \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[edit]
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[edit]
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.
[show]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
∇ 2 Φ [ x → , t ] = 4 π G ρ [ x → , t ] {\displaystyle \nabla ^{2}\Phi \left[{\vec {x}},t\right]=4\pi G\rho \left[{\vec {x}},t\right]}
where ρ is the mass density. The orbit of a free-falling particle satisfies
x → ¨ [ t ] = − ∇ Φ [ x → [ t ] , t ] . {\displaystyle {\ddot {\vec {x}}}[t]=-\nabla \Phi \left[{\vec {x}}[t],t\right]\,.}
In tensor notation, these become
Φ , i i = 4 π G ρ d 2 x i d t 2 = − Φ , i . {\displaystyle {\begin{aligned}\Phi _{,ii}&=4\pi G\rho \\{\frac {d^{2}x^{i}}{dt^{2}}}&=-\Phi _{,i}\,.\end{aligned}}}
In general relativity, these equations are replaced by the Einstein field equations in the trace-reversed form
R μ ν = K ( T μ ν − 1 2 T g μ ν ) {\displaystyle R_{\mu \nu }=K\left(T_{\mu \nu }-{\tfrac {1}{2}}Tg_{\mu \nu }\right)}
for some constant, K, and the geodesic equation
d 2 x α d τ 2 = − Γ β γ α d x β d τ d x γ d τ . {\displaystyle {\frac {d^{2}x^{\alpha }}{d\tau ^{2}}}=-\Gamma _{\beta \gamma }^{\alpha }{\frac {dx^{\beta }}{d\tau }}{\frac {dx^{\gamma }}{d\tau }}\,.}
To see how the latter reduces to the former, we assume that the test particle's velocity is approximately zero
d x β d τ ≈ ( d t d τ , 0 , 0 , 0 ) {\displaystyle {\frac {dx^{\beta }}{d\tau }}\approx \left({\frac {dt}{d\tau }},0,0,0\right)}
and thus
d d t ( d t d τ ) ≈ 0 {\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
d 2 x i d t 2 ≈ − Γ 00 i {\displaystyle {\frac {d^{2}x^{i}}{dt^{2}}}\approx -\Gamma _{00}^{i}}
where two factors of dt/dτ have been divided out. This will reduce to its Newtonian counterpart, provided
Φ , i ≈ Γ 00 i = 1 2 g i α ( g α 0 , 0 + g 0 α , 0 − g 00 , α ) . {\displaystyle \Phi _{,i}\approx \Gamma _{00}^{i}={\tfrac {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
2 Φ , i ≈ g i j ( − g 00 , j ) ≈ − g 00 , i {\displaystyle 2\Phi _{,i}\approx g^{ij}\left(-g_{00,j}\right)\approx -g_{00,i}\,}
which is satisfied by letting
g 00 ≈ − c 2 − 2 Φ . {\displaystyle g_{00}\approx -c^{2}-2\Phi \,.}
Turning to the Einstein equations, we only need the time-time component
R 00 = K ( T 00 − 1 2 T g 00 ) {\displaystyle R_{00}=K\left(T_{00}-{\tfrac {1}{2}}Tg_{00}\right)}
the low speed and static field assumptions imply that
T μ ν ≈ d i a g ( T 00 , 0 , 0 , 0 ) ≈ d i a g ( ρ c 4 , 0 , 0 , 0 ) . {\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
T = g α β T α β ≈ g 00 T 00 ≈ − 1 c 2 ρ c 4 = − ρ c 2 {\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
K ( T 00 − 1 2 T g 00 ) ≈ K ( ρ c 4 − 1 2 ( − ρ c 2 ) ( − c 2 ) ) = 1 2 K ρ c 4 . {\displaystyle K\left(T_{00}-{\tfrac {1}{2}}Tg_{00}\right)\approx K\left(\rho c^{4}-{\tfrac {1}{2}}\left(-\rho c^{2}\right)\left(-c^{2}\right)\right)={\tfrac {1}{2}}K\rho c^{4}\,.}
From the definition of the Ricci tensor
R 00 = Γ 00 , ρ ρ − Γ ρ 0 , 0 ρ + Γ ρ λ ρ Γ 00 λ − Γ 0 λ ρ Γ ρ 0 λ . {\displaystyle R_{00}=\Gamma _{00,\rho }^{\rho }-\Gamma _{\rho 0,0}^{\rho }+\Gamma _{\rho \lambda }^{\rho }\Gamma _{00}^{\lambda }-\Gamma _{0\lambda }^{\rho }\Gamma _{\rho 0}^{\lambda }.}
Our simplifying assumptions make the squares of Γ disappear together with the time derivatives
R 00 ≈ Γ 00 , i i . {\displaystyle R_{00}\approx \Gamma _{00,i}^{i}\,.}
Combining the above equations together
Φ , i i ≈ Γ 00 , i i ≈ R 00 = K ( T 00 − 1 2 T g 00 ) ≈ 1 2 K ρ c 4 {\displaystyle \Phi _{,ii}\approx \Gamma _{00,i}^{i}\approx R_{00}=K\left(T_{00}-{\tfrac {1}{2}}Tg_{00}\right)\approx {\tfrac {1}{2}}K\rho c^{4}}
which reduces to the Newtonian field equation provided
1 2 K ρ c 4 = 4 π G ρ {\displaystyle {\tfrac {1}{2}}K\rho c^{4}=4\pi G\rho \,}
which will occur if
K = 8 π G c 4 . {\displaystyle K={\frac {8\pi G}{c^{4}}}\,.}
Vacuum field equations[edit]
A Swiss commemorative coin from 1979, showing the vacuum field equations with zero cosmological constant (top).
If the energy-momentum tensor Tμν is zero in the region under consideration, then the field equations are also referred to as the vacuum field equations. By setting Tμν = 0 in the trace-reversed field equations, the vacuum equations can be written as
R μ ν = 0 . {\displaystyle R_{\mu \nu }=0\,.}
In the case of nonzero cosmological constant, the equations are
R μ ν = Λ D 2 − 1 g μ ν . {\displaystyle R_{\mu \nu }={\frac {\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, Rμν = 0, are referred to as Ricci-flat manifolds and manifolds with a Ricci tensor proportional to the metric as Einstein manifolds.
Einstein–Maxwell equations[edit]
See also: Maxwell's equations in curved spacetime
If the energy-momentum tensor Tμν is that of an electromagnetic field in free space, i.e. if the electromagnetic stress–energy tensor
T α β = − 1 μ 0 ( F α ψ F ψ β + 1 4 g α β F ψ τ F ψ τ ) {\displaystyle T^{\alpha \beta }=\,-{\frac {1}{\mu _{0}}}\left({F^{\alpha }}^{\psi }{F_{\psi }}^{\beta }+{\tfrac {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):
R α β − 1 2 R g α β + Λ g α β = 8 π G c 4 μ 0 ( F α ψ F ψ β + 1 4 g α β F ψ τ F ψ τ ) . {\displaystyle R^{\alpha \beta }-{\tfrac {1}{2}}Rg^{\alpha \beta }+\Lambda g^{\alpha \beta }={\frac {8\pi G}{c^{4}\mu _{0}}}\left({F^{\alpha }}^{\psi }{F_{\psi }}^{\beta }+{\tfrac {1}{4}}g^{\alpha \beta }F_{\psi \tau }F^{\psi \tau }\right).}
Additionally, the covariant Maxwell equations are also applicable in free space:
F α β ; β = 0 F [ α β ; γ ] = 1 3 ( F α β ; γ + F β γ ; α + F γ α ; β ) = 1 3 ( F α β , γ + F β γ , α + F γ α , β ) = 0. {\displaystyle {\begin{aligned}{F^{\alpha \beta }}_{;\beta }&=0\\F_{[\alpha \beta ;\gamma ]}&={\tfrac {1}{3}}\left(F_{\alpha \beta ;\gamma }+F_{\beta \gamma ;\alpha }+F_{\gamma \alpha ;\beta }\right)={\tfrac {1}{3}}\left(F_{\alpha \beta ,\gamma }+F_{\beta \gamma ,\alpha }+F_{\gamma \alpha ,\beta }\right)=0.\end{aligned}}}
where the semicolon represents a covariant derivative, and the brackets denote anti-symmetrization. The first equation asserts that the 4-divergence of the two-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 Aα such that
F α β = A α ; β − A β ; α = A α , β − A β , α {\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.[14] However, there are global solutions of the equation which may lack a globally defined potential.[15]
Solutions[edit]
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 numerous cases where the field equations have been solved completely, and those are called exact solutions.[6]
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.[16] In this approach, the Einstein field equations are reduced to a set of coupled, nonlinear, ordinary differential equations. As discussed by Hsu and Wainwright,[17] 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 [18] and Kohli and Haslam.[19]
The linearized EFE[edit]
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, with terms that are quadratic in or higher powers of the deviation being ignored. This linearization procedure can be used to investigate the phenomena of gravitational radiation.
Polynomial form[edit]
One might think that EFE are non-polynomial since they contain the inverse of the metric tensor. However, the equations can be arranged so that they contain only the metric tensor and not its inverse. First, the determinant of the metric in 4 dimensions can be written:
det ( g ) = 1 24 ε α β γ δ ε κ λ μ ν g α κ g β λ g γ μ g δ ν {\displaystyle \det(g)={\tfrac {1}{24}}\varepsilon ^{\alpha \beta \gamma \delta }\varepsilon ^{\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:
g α κ = 1 6 ε α β γ δ ε κ λ μ ν g β λ g γ μ g δ ν det ( g ) . {\displaystyle g^{\alpha \kappa }={\frac {{\tfrac {1}{6}}\varepsilon ^{\alpha \beta \gamma \delta }\varepsilon ^{\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 det(g) until there are none left in 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.[20]
See also[edit]
Einstein–Hilbert action
Equivalence principle
Exact solutions in general relativity
General relativity resources
History of general relativity
Hamilton–Jacobi–Einstein equation
Mathematics of general relativity
Ricci calculus
Notes[edit]
^ Jump up to: a b Einstein, Albert (1916). "The Foundation of the General Theory of Relativity". Annalen der Physik. 354 (7): 769. Bibcode:1916AnP...354..769E. doi:10.1002/andp.19163540702. Archived from the original (PDF) on 2012-02-06.
Jump up ^ Einstein, Albert (November 25, 1915). "Die Feldgleichungen der Gravitation". Sitzungsberichte der Preussischen Akademie der Wissenschaften zu Berlin: 844–847. Retrieved 2017-08-21.
Jump up ^ Misner, Charles W.; Thorne, Kip S.; Wheeler, John Archibald (1973). Gravitation. San Francisco: W. H. Freeman. ch. 34, p. 916. ISBN 978-0-7167-0344-0.
Jump up ^ Carroll, Sean (2004). Spacetime and Geometry – An Introduction to General Relativity. pp. 151–159. ISBN 0-8053-8732-3.
Jump up ^ Grøn, Øyvind; Hervik, Sigbjorn (2007). Einstein's General Theory of Relativity: With Modern Applications in Cosmology (illustrated ed.). Springer Science & Business Media. p. 180. ISBN 978-0-387-69200-5.
^ Jump up to: a b Stephani, Hans; Kramer, D.; MacCallum, M.; Hoenselaers, C.; Herlt, E. (2003). Exact Solutions of Einstein's Field Equations. Cambridge University Press. ISBN 0-521-46136-7.
Jump up ^ Weinberg, Steven (1993). Dreams of a Final Theory: the search for the fundamental laws of nature. Vintage Press. pp. 107, 233. ISBN 0-09-922391-0.
Jump up ^ Misner, Thorne & Wheeler 1973
Jump up ^ Weinberg 1972
Jump up ^ Peacock 1994
Jump up ^ Gamow, George (April 28, 1970). My World Line : An Informal Autobiography. Viking Adult. ISBN 0-670-50376-2. Retrieved 2007-03-14.
Jump up ^ Wahl, Nicolle (2005-11-22). "Was Einstein's 'biggest blunder' a stellar success?". Archived from the original on 2007-03-07. Retrieved 2007-03-14.
Jump up ^ Turner, Michael S. (May 2001). "Making Sense of the New Cosmology". Int. J. Mod. Phys. A. 17 (S1): 180–196. arXiv:astro-ph/0202008  . Bibcode:2002IJMPA..17S.180T. doi:10.1142/S0217751X02013113.
Jump up ^ Brown, Harvey (2005). Physical Relativity. Oxford University Press. p. 164. ISBN 978-0-19-927583-0.
Jump up ^ Trautman, Andrzej (1977). "Solutions of the Maxwell and Yang–Mills equations associated with Hopf fibrings". International Journal of Theoretical Physics. 16 (9): 561–565. Bibcode:1977IJTP...16..561T. doi:10.1007/BF01811088. .
Jump up ^ Ellis, G. F. R.; MacCallum, M. (1969). "A class of homogeneous cosmological models". Comm. Math. Phys. 12 (2): 108–141. Bibcode:1969CMaPh..12..108E. doi:10.1007/BF01645908.
Jump up ^ Hsu, L.; Wainwright, J (1986). "Self-similar spatially homogeneous cosmologies: orthogonal perfect fluid and vacuum solutions". Class. Quantum Grav. 3: 1105–1124.
Jump up ^ LeBlanc, V. G. (1997). "Asymptotic states of magnetic Bianchi I cosmologies". Class. Quantum Grav. 14: 2281.
Jump up ^ Kohli, Ikjyot Singh; Haslam, Michael C. (2013). "Dynamical systems approach to a Bianchi type I viscous magnetohydrodynamic model". Phys. Rev. D. 88: 063518.
Jump up ^ Katanaev, M. O. (July 7, 2005). "Polynomial form of the Hilbert–Einstein action". Gen. Rel. Grav. 38: 1233–1240. arXiv:gr-qc/0507026  . Bibcode:2006GReGr..38.1233K. doi:10.1007/s10714-006-0310-5.
References[edit]
See General relativity resources.
Misner, Charles; Thorne, Kip S.; Wheeler, John Archibald (1973), Gravitation, San Francisco: W. H. Freeman, p. 501ff, ISBN 0-7167-0344-0
Weinberg, Steven (1972), Gravitation and Cosmology, John Wiley & Sons, ISBN 0-471-92567-5
Peacock, John A. (1994), Cosmological Physics, Cambridge University Press, ISBN 978-0521410724
External links[edit]
Wikibooks has a book on the topic of: General Relativity
Wikiversity has learning resources about General Relativity
Hazewinkel, Michiel, ed. (2001) [1994], "Einstein equations", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
Caltech Tutorial on Relativity — A simple introduction to Einstein's Field Equations.
The Meaning of Einstein's Equation — An explanation of Einstein's field equation, its derivation, and some of its consequences
Video Lecture on Einstein's Field Equations by MIT Physics Professor Edmund Bertschinger.
Arch and scaffold: How Einstein found his field equations Physics Today November 2015, History of the Development of the Field Equations
The Einstein field equation on the wall of the Museum Boerhaave in downtown Leiden
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values: {\displaystyle g_{00}\approx -c^{2}-2\Phi \,.}

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