일반 상대성 이론의 대안

이 기사의 사실적 정확성은 논란의 여지가 있다. 관련 토론은 토크 페이지에서 확인할 수 있습니다. 문제가 있는 진술이 신뢰할 수 있는 출처인지 확인하시기 바랍니다. (2016년 2월) (이 템플릿 메시지 삭제 방법 및 시기 확인)

일반 상대성 이론의 대안은 아인슈타인의 일반 상대성 이론과 경쟁하여 중력의 현상을 설명하려는 물리 이론이다.이상적인 ^[1]중력 이론을 구축하기 위한 많은 다양한 시도가 있었다.

이러한 시도는 범위에 따라 크게 4가지 범주로 나눌 수 있습니다.이 기사에서는 양자역학이나 힘통일을 수반하지 않는 일반상대론에 대한 간단한 대안이 논의된다.양자역학의 원리를 이용하여 이론을 구성하려는 다른 이론들은 양자화된 중력 이론으로 알려져 있다.셋째, 중력과 다른 힘을 동시에 설명하려는 이론들이 있다; 이것들은 고전적인 통합장 이론으로 알려져 있다.마지막으로, 가장 야심찬 이론들은 중력을 양자역학적 용어로 놓고 힘을 통합하려고 시도한다; 이것들은 모든 것의 이론이라고 불린다.

일반상대성이론에 대한 이러한 대안들 중 어느 것도 널리 받아들여지지 않았다.일반상대성이론은 지금까지의 모든 관측과 일관성을 유지하면서 많은 테스트를 ^[2]견뎌왔다.이와는 대조적으로, 많은 초기 대안들은 확실히 반증되었다.하지만, 중력 이론 중 일부는 소수의 물리학자들에 의해 지지를 받고 있고, 그 주제는 이론 물리학에서 집중적인 연구의 주제로 남아있다.

일반상대성이론을 통한 중력 이론의 역사

17세기에 출판되었을 때, 아이작 뉴턴의 중력 이론은 가장 정확한 중력 이론이었다.그 이후로 많은 대안이 제시되었다.1915년 일반상대성이론의 공식화 이전의 이론은 중력 이론의 역사에서 논의되었다.

일반상대성이론

이 이론은^[3][4] 현재 우리가 "일반 상대성 이론"이라고 부르는 것입니다(비교를 위해 여기에 포함됨).민코프스키 메트릭을 완전히 폐기하면 아인슈타인은 다음을 얻는다.

${\displaystyle\int ds=0,}$

${\displaystyle {ds}^{2}=g_{\mu \nu },g^{\mu },g^{\nu },}$

${\displaystyle R_{\mu \nu } = scfrac {8\pi G} {c^{4}} \left(T_{\mu \nu }-{\frac {1}{2}}g_{\mu \nu }T\right},$

쓸 수도 있다

$\displaystyle T^{\mu \nu }={c^{4} \over 8\pi G}\left(R^{\mu \nu }-{\frac {1}{2}}g^{\mu \nu }R\right},}$

아인슈타인이 위의 마지막 방정식을 제시하기 5일 전에 힐버트는 거의 동일한 방정식을 포함하는 논문을 제출했다.일반 상대성 우선 순위 분쟁을 참조하십시오.힐버트는 아인슈타인을 정확히 말한 최초의 사람이다.일반 상대성 이론에 대한 힐버트 작용, 즉,

${\displaystyle S={c^{4}\over 16\pi G}\int R{\sqrt {-g}}\d^{4}x+S_{m},}$

$여기$서$$G({$displaystyle$ G$}$는 Newton의 중력 상수, $R{\displaystyle }$ $μμ{\mu}^{~\mu},}$는 $Ricci{\displaystyle }$ 곡률, g $=$ ${\displaystyle det(g_{}}$ $)$ {$display$ g=\$det(g_{\mu \nu})$ $및{\displaystyle {}_{}}$ $S_style$ S_m {\$m}$입니다.

일반상대성이론은 텐서 이론이고 방정식은 모두 텐서를 포함하고 있다.반면에 노드스트롬의 이론은 중력장이 스칼라이기 때문에 스칼라 이론이다.제안된 다른 대안으로는 일반 상대성 이론 외에 스칼라 필드를 포함하는 스칼라-텐서 이론과 벡터 필드를 포함하는 다른 변종들이 최근에 개발되었다.

동기

일반상대성이론 이후 일반상대성이론 이전에 개발된 이론을 개선하거나 일반상대성이론 자체를 개선하려는 시도가 있었다.일반 상대성 이론에 스핀을 추가하는 등 많은 다른 전략들이 시도되었는데, 일반 상대성 이론과 같은 측정 기준과 우주의 팽창에 대해 정적인 시공간을 결합하여, 또 다른 매개변수를 추가함으로써 추가적인 자유를 얻습니다.적어도 하나의 이론은 특이점이 없는 일반 상대성 이론의 대안을 개발하려는 욕망에 의해 동기 부여되었다.

실험 테스트는 이론과 함께 개선되었다.일반상대성이론 이후에 개발된 많은 다른 전략들은 포기되었고, 어떤 테스트가 일반상대성이론과 불일치를 보일 때 이론이 준비될 수 있도록 살아남은 이론들의 보다 일반적인 형태를 개발하려는 노력이 있었다.

1980년대에 이르러서는 실험의 정확도가 높아지면서 일반상대성이론이 확인되었고, 특별한 경우로 일반상대성이론을 포함하는 경우를 제외하고는 경쟁자가 남아있지 않았다.게다가, 그 직후, 이론가들은 유망해 보이기 시작했지만, 그 이후로 인기를 잃었다.1980년대 중반에는 몇 미터 범위에서 작용하는 다섯 번째 힘(또는 경우에 따라서는 다섯 번째 힘, 여섯 번째 힘, 일곱 번째 힘)의 추가에 의해 중력이 변경되고 있다는 몇 가지 실험이 있었다.후속 실험에서 이러한 것들이 제거되었다.

보다 최근의 대안 이론의 동기는 거의 모든 우주론이며, "인플레이션", "암흑 물질", "암흑 에너지"와 관련되거나 대체된다.파이오니어 변칙에 대한 조사는 일반 상대성 이론의 대안에 대한 대중의 관심을 다시 불러일으켰다.

이 문서의 표기법

$(\$c\;)는 빛의 속도$,$ $G{\displaystyle }$(\$displaystyle$ G$\;)$는 중력 상수입니다."기하 변수"는 사용되지 않습니다.

라틴 지수는 1에서 3까지, 그리스 지수는 0에서 3까지입니다.아인슈타인 요약 규칙이 사용됩니다.

${\displaystyle {\mu }_{}}$ μ$ν$ {$displaystyle$ \$eta _$ { \ $mu$ \ nu$$} \ ; }는 민코프스키 $메트릭$입니다.${\displaystyle g_{}}$ $μ$ ${\displaystyle \{\backslash {}_{}}$ { \ $displaystyle$ g$_$ { \ $mu$ \ nu $}$ \ ; }는$$ 텐서이며, 일반적으로 메트릭 텐서입니다.이것들은 시그니처(-,+,+,+)를 가집니다.

편미분은${\displaystyle {\mathrm{}}_{}}$ ${\$ \ $displaystyle$\$partial$ _ { \ $mu }$\ varphi$$ \ ; ${\displaystyle {,}_{}}$ 、${\displaystyle {}_{}}{\displaystyle {\mu }_{}}$ \ $displaystyle$\ $varphi$ $_$$${ , \ $mu }$ \ ;$$${\displaystyle {\mathrm{\mu <img\; alt="\{\backslash displaystyle\; \backslash nabla\; \_\{\backslash mu\; \}\backslash varphi\; \backslash ;\}"\; aria-hidden="true"\; class="mwe-math-fallback-image-inline"\; src="https://wikimedia.org/api/rest\_v1/media/math/render/svg/44a73a794945ec7662231668823b0b69c1b1695b"\; style="vertical-align:\; -1.005ex;\; width:5.325ex;\; height:2.843ex;">}}_{}}$ ${\$ $\{\backslash {\displaystyle {\mathrm{}}_{}}$ ${\displaystyle {}_{}}$ partial 、 \ $displaystyle$\$nabla$ _ { \ $mu$} \ $varphi$$$\ ;${\displaystyle {}_{}}${\ {\ {\ ${\displaystyle \{\backslash {}_{}}$ {\ {\ {\ ${\$ {\ {\ {\ {\ {\ μ {\ {\ {\ {\ {\ {\ {\ {\ {\ {\ {\ {\

이론의 분류

중력 이론은 느슨하게 여러 범주로 분류될 수 있다.여기서 설명하는 대부분의 이론은 다음과 같습니다.

• '동작' (작동 개념에 기초한 변이 원리인 최소 작용의 원칙 참조)
• 라그랑주 밀도
• 미터법

이론이 중력에 대해 라그랑지안 밀도를 갖는 경우$,$ $예$를 들어$$L(\$displaystyle$ L$$$,)$ $동작$의 중력 부분은 다음과 같습니다.

${\displaystyle S}$ $=$ ${\displaystyle \int \sqrt{}L}$ - ${\displaystyle \sqrt{g}}$ ${\displaystyle {\mathrm{}}^{}}$ 4$$ {\$displaystyle$ S=\$int$ L${\sqrt${-$g}},\mathrm {d}^{4}x$

이 방정식에서는 데카르트 좌표를 사용할 때 공간 무한대에서 g $=$ -$$(\$displaystyle$ g=-$1,)$을 $갖는{\displaystyle }$ 것이 일반적이지만 필수는 아니다.예를 들어 아인슈타인은...힐베르트 동작은

$\displaystyle L,\propto \,R$

여기서 R은 스칼라 곡률로 공간 곡률의 측정값입니다.

이 기사에 기술된 거의 모든 이론에는 작용이 있다.에너지, 운동량 및 각운동량의 필수 보존 법칙이 자동으로 통합되도록 보장하는 가장 효율적인 방법입니다. 그러나 이러한 보존 법칙을 위반하는 행동을 구축하는 것은 쉽습니다.표준 방법은 필요한 보존 법칙을 가진 시스템을 구축하는 또 다른 방법을 제공하지만,^[5] 이 방법은 구현하기가 더 번거롭습니다.1983년판 MOND는 액션이 없었습니다.

몇몇 이론들은 작용은 있지만 라그랑주 밀도는 아니다.좋은 예로는 ^[6]Whitehead를 들 수 있습니다. Whitehead는 여기서의 액션을 non-local이라고 부릅니다.

중력 이론은 다음 두 조건이 성립하는 수학적 표현을 제공할 수 있는 경우에만 "측정학 이론"이다.
조건 1: 시그니처(-, +, +, +, +, +)의 대칭 메트릭 ${\displaystyle {}_{}}$ ${\displaystyle g_{}}$ ${\displaystyle {\mu }_{}}$ ${\displaystyle †{}_{}}$ {\$displaystyle g_{\mu$ \nu$$},}가$$ 존재하며, 이는 특수 및 일반 상대성 이론의 통상적인 방법으로 적절한 길이 및 적절한 시간 측정을 지배한다.

${\displaystyle {d\nu}^{2}=-g_{\mu \nu },contrace^{\mu },contrace^{\nu },}$

여기서 $\mu {\displaystyle }${\$displaystyle \mu}$ 및$$ ${\$ {\$displaystyle \nu$ 인덱스에 대한 합계가 있습니다.
조건 2: 중력에 의해 작용하는 응력 물질과 장은 다음 식에 따라 반응한다.

${\displaystyle 0=\nu }T^{\mu \nu }={\nu }}_{,\nu }+\Gamma _{\sigma \nu }^{\nu }+\Gamma _{\nu } ^{\nu } } ^{\t}$

${\displaystyle {여기}^{}}$서 T μ${\displaystyle {}^{}}{\displaystyle {\delta }^{}}$ {\$displaystyle$ T$^{\mu \nu }$는 모든 물질 및 비중력장에 대한 응력-에너지 텐서이고,$${\$nu$}는$$ 메트릭에 대한 공변 도함수이고 $\delta {\displaystyle {\mathrm{}}_{}^{}}$ ${\displaystyle {}_{}^{}}$ ${\displaystyle$ $\$$Gamma {\s}$는 $α$^{\nu }입니다.응력-에너지 텐서는 에너지 조건도 충족해야 한다.

메트릭 이론에는 다음과 같은 것이 있습니다(단순한 것부터 가장 복잡한 것까지).

• 스칼라 필드 이론(컨포멀리 플랫 이론 및 컨포멀리 플랫 스페이스 슬라이스를 가진 계층화 이론 포함)
  • 베르크만
  • 콜먼
  • 아인슈타인 (1912년)
  • 아인슈타인-포커 이론
  • 리 라이트맨 니
  • 리틀우드
  • 니
  • 노드스트롬의 중력 이론(최초 개발되는 중력 측정 이론)
  • 페이지 – 탭
  • 파파페트로우
  • 로젠(1971년)
  • 휘트로-모르두흐
  • 일마즈 중력 이론(이론에서 사건의 지평을 제거하려고 시도)
• 준선형 이론(선형 고정 게이지 포함)
  • 볼리니-지암바기-티옴노
  • 디저-로랑
  • 화이트헤드의 중력 이론(지체 전위만 사용)
• 텐서 이론
  • 아인슈타인의 일반 상대성 이론
  • 4차 중력(라그랑지안이 리만 곡률 텐서의 2차 수축에 의존할 수 있음)
  • f(R) 중력(라그랑지안이 리치 스칼라의 더 높은 힘에 의존할 수 있게 함)
  • 가우스-보네 중력
  • Lovelock 중력 이론(라그랑지안이 리만 곡률 텐서의 고차 수축에 의존하도록 허용)
  • 무한 미분 중력
• 스칼라-텐서 이론
  • 베켄슈타인
  • 베르그만-웨거
  • 브랜스-딕케 이론(마하 원리를 더 잘 적용하기 위한 일반 상대성 이론의 가장 잘 알려진 대안)
  • 조던
  • 노르트베데트
  • 티리
  • 카멜레온
  • 가압기
• 벡터-텐서 이론
  • 헬링스-노르트베트
  • 윌-노르트베트
• 바이메트릭 이론
  • 라이트맨리
  • 라스타루
  • 로젠(1975)
• 기타 메트릭 이론

(아래의 '현대 이론' 섹션 참조)

비금속 이론에는 다음이 포함된다.

• 벨인판테-스위하르트
• 아인슈타인-카르탄 이론(회전 궤도 각운동량 교환을 다루려는 의도)
• 쿠스탄헤이모 (1967년)
• 텔레패럴리즘
• 게이지 이론 중력

여기서^[6] 마하 원리에 대한 단어는 이 이론들 중 몇 가지는 마하 원리에 의존하고 많은 이론들이 지나가는 동안 그것을 언급하기 때문에 적절하다(예: 아인슈타인-그로스만,^[7] 브랜스-딕케^[8].마하의 원리는 뉴턴과 아인슈타인 사이의 중간집합이라고 생각할 수 있다.이 방법은 ^[9]다음과 같습니다.

• 뉴턴: 절대 시공간이야.
• 마하: 기준 프레임은 우주의 물질 분포에서 비롯됩니다.
• 아인슈타인:참조 프레임이 없습니다.

지금까지의 모든 실험 증거는 마하의 원리가 틀렸다고 지적하고 있지만 완전히 ^{[citation needed]}배제된 것은 아니다.

1917년부터 1980년대까지의 이론

이 대분류는 일반상대성이론 이후에 발표된 일반상대성이론에 대한 대안과 "암흑물질" 가설로 이어진 은하 회전 관측 이전에 발표된 내용을 포함한다.여기서 검토되는 것은 다음과 같습니다(Will^[12][13] Lang 참조^[10][11]).

1917년부터 1980년대까지의 이론들.

발행년도	작성자	이론명	이론형
1922년[6]	앨프리드 노스 화이트헤드	화이트헤드의 중력 이론	준선형
1922,[14] 1923[15]	엘리 카르탕	아인슈타인-카르탄 이론	비메트릭
1939년[16]	마르쿠스 피에르츠, 볼프강 파울리
1943년[17]	조지 데이비드 버크호프
1948년[18]	에드워드 아서 밀른	운동 상대성 이론
1948년[19]	이브 티리
1954년[20][21]	아킬레스 파파페트루		[ Scalar ]필드
1953년[22]	더들리 E.리틀우드		[ Scalar ]필드
1955년[23]	파스쿠알 요르단
1956년[24]	오토 베르그만		[ Scalar ]필드
1957년[25][26]	프레데릭 벨인판테, 제임스 C스와하르트
1958,[27] 1973[28]	후세인 일마즈	일마즈 중력 이론
1961년[8]	칼 H. 브랜스, 로버트 H. 딕	브랜스-딕케 이론	스칼라텐서
1960,[29] 1965[30]	Gerald James Whitrow, G.E. Morduch		[ Scalar ]필드
1966년[31]	폴 쿠스탄헤이모[de]
1967년[32]	폴 쿠스탄헤이모[데], V. S. 누오티오
1968년[33]	스탠리 디저, B. E. 로랑		준선형
1968년[34]	C. 페이지, B. O. J. Tupper		[ Scalar ]필드
1968년[35]	피터 버그만		스칼라텐서
1970년[36]	C. G. 볼리니, J. 지암바기, J. 티옴노		준선형
1970년[37]	케네스 노르트베트
1970년[38]	로버트 5세웨고너		스칼라텐서
1971년[39]	네이선 로젠		[ Scalar ]필드
1975년[40]	네이선 로젠		바이메트릭
1972년[41],[11] 1973년	니웨이투		[ Scalar ]필드
1972년[42]	클리포드 마틴 윌, 케네스 노르트베트		벡터-텐서
1973년[43]	로널드 헬링스, 케네스 노르트베트		벡터-텐서
1973년[44]	앨런 라이트먼, 데이비드 L. 리		[ Scalar ]필드
1974년[45]	데이비드 L. 리, 앨런 라이트먼, 니웨이투
1977년[46]	야콥 베켄슈타인		스칼라텐서
1978년[47]	B. M. 바커		스칼라텐서
1979년[48]	P. 라스톨		바이메트릭

이 이론들은 특별히 언급되지 않는 한 우주 상수나 스칼라 또는 벡터 잠재력 없이 여기에 제시된다. 이는 초신성 우주론 프로젝트와 High-Z 초신성 탐색 팀이 초신성 관측 전에 이들 중 하나 또는 둘의 필요성이 인식되지 않았기 때문이다.이론에 우주 상수 또는 정수를 추가하는 방법은 현대 이론에서 논의됩니다(아인슈타인도 참조).힐베르트 액션).

스칼라 필드 이론

노르드스트롬의^[49][50] 스칼라장 이론은 이미 논의되었다.Littlewood,^[22][24] Bergman, Yilmaz,^[27] Whitrow와 Morduch^[29][30], Page와 Tupper는^[34] 페이지와 Tupper가 제시한 일반적인 공식을 따릅니다.

Nordström을 ^[50]제외한 이 모든 것을 논의한 페이지와 ^[34]Tupper에 따르면 일반적인 스칼라 장 이론은 최소 작용의 원리에서 비롯된다.

$\displaystyle \int f\leftfrac {varphi }{c^{2}}\right),ds=0}$

스칼라 필드가 있는 경우

$\displaystyle \varphi = scfrac {GM}{r}}$

및 c는 $"\displaystyle\varphi$에 의존하거나 의존하지 않을 수 있습니다.

노드스트롬에서는^[49]

$\displaystyle f(\varphi /c^{2}=\expair\varphi /c^{2}),\qquad c=c_{\infty }}$

리틀우드와^{[22] [24]}버그만에서

${{displaystyle f\left f\frac {varphi }{c^{2}\right}=\exp \frac {\frac {(c/\varphi ^{2}}{2}}-{qquad=c_{\infty}}},$

휘트로와 ^[29]모듀크에서는

${\displaystyle f\left f\frac {c^{2}}\right}=1,\qquad c^{2}=c_{\infty }^{2}-2\varphi,}$

휘트로와 ^[30]모듀크에서는

${\displaystyle f\left f\frac {c^{2}}\right}=\exp \left fac {\frac {\frac {c^{2}=c_{\infty }^{2}-2\varphi },\exp \f \fac {\fac}\fac {\frac {\f}\f}\fright}\frac {\f}\frac {\f},\f},\f},\frac c^{c^{c^{c^{c^{c$

페이지 및 ^[34]터퍼에서

$({displaystyle f\left\frac {c^{2}}\right) = frac {varphi } {c^{2} +\alpha \left\frac {c^{2} } {right} {qquad {frac {c_{\infty} } = {c2} } = {c} } } } }$

페이지와^[34] 터퍼는 Yilmaz의^[27] 이론을${\displaystyle }$α $=$ - ${\displaystyle 7\mathrm{}}$ /$$2({$displaystyle$ \alpha =-$7/2$의 2차 차수와 일치시킵니다.

c가 일정할 때 빛의 중력편향은 0이어야 한다.가변 c와 빛의 0편향이 모두 실험과 상충된다는 점을 고려하면 성공적인 스칼라 중력 이론의 가능성은 매우 희박해 보입니다.또, 스칼라 이론의 파라메타를 빛의 편향이 맞도록 조정하면, 중력 적색 편이 어긋나기 쉽다.

니는^[11] 몇 가지 이론을 요약하고 두 가지를 더 만들었다.우선 기존의 특수상대성이론 시공좌표와 비중력장과 작용하여 스칼라장을 생성한다.이 스칼라 필드는 다른 모든 필드와 함께 동작하여 메트릭을 생성합니다.

액션은 다음과 같습니다.

${\displaystyle S={1\over 16\pi G}\int d^{4}xsecrt {-g}}L_{\varphi}+S_{m}}$

$\displaystyle L_{\varphi }=\varphi R-2g^{\mu \nu },\display_{\mu }\varphi },\display_{\nu }\varphi }$

Misner 등은 이것을 ^[51]"$\displaystyle\varphi$ R$"$ 없이 제시한다.$(\$은 문제 조치입니다.

$\displaystyle \Varphi = 4\pi T^{\mu \nu }\left[\eta _{\mu \nu }e^{-2\varphi }+e^{-2\varphi }\right,\tnu }\left$

t는 세계시 좌표입니다.이 이론은 자기 정합적이고 완전하다.하지만 우주를 통과하는 태양계의 움직임은 실험과 심각한 불일치로 이어진다.

Ni의^[11] 두 번째 이론에서는 다음과 같이 메트릭과 관련된 두 개의 ${\displaystyle \mathrm{임의}}$ $함수{\displaystyle }$ f${\displaystyle (}$ ${\displaystyle )}$) {${\displaystyle \mathrm{displaystyle}}$ f$(\varphi)}$ 및$$ $k{\displaystyle (}$ ${\displaystyle )}$) {$displaystyle$k(\$varphi)}$가 있습니다.

${\displaystyle ds^{2}=e^{2}-e^{2f(\varphi)}\left[blight]$

$\displaystyle \eta ^{\mu \nu }\display _{\mu }\display _{\nu }\varphi = 4\pi \rho ^{*}k(\varphi)}$

Ni는^[11] Rosen이 메트릭과 관련된 두 개의 스칼라 $필드{\displaystyle }$ $"\displaystyle \varphi"$ 및$$ $"\displaystyle \psi"$를 가지고 있다고 인용합니다^[39].

$({displaystyle ds^{2}=\varphi^{2},dt^{2}-\psi^{2}\left[display^{2}+dz^{2}\right])$

파파페트로우에서^[20] 라그랑지안의 중력 부분은 다음과 같다.

$\displaystyle L_{\varphi }=e^{\varphi }\lefts\tfrac {1}{2}}e^{-\varphi},\displays _{\alpha }\varphi +{\tfrac {3}{\varphi },\displays _{\varphi}$

파파페트루에는^[21] 두 번째 스칼라 필드$\delta {\displaystyle }$(\$displaystyle\chi$가 있습니다. 라그랑지안의 중력 부분은 다음과 같습니다.

$\displaystyle L_{\varphi}=e^{\frac {1}{2}(3\varphi +\chi)}\lefts{\tfrac {1}{2},\flash _{\alphi }\varphi },\frac _{\fi } - varphi }$

바이메트릭 이론

바이메트릭 이론은 정규 텐서 메트릭과 민코프스키 메트릭(또는 일정한 곡률 메트릭)을 모두 포함하며, 다른 스칼라 또는 벡터 필드를 포함할 수 있다.

로젠^[52](1975) 바이메트릭 이론 작용:

${\displaystyle S={1 \over 64 \pi G}\int d^{4}x,{\displayrt {-\mu \nu }}\eta ^{\alpha \nu }g^{\display }g^{\alpha \mu }g }g^{\display }g }g }{{\text } {\text } {\text } } {\text } {\text } {{{{{{\frcrcrcS_{m}}$

$\Displaystyle \Box _{\eta }g_{\mu \nu }-g^{\alpha \nu }g_{\nu \sqrt {g/\eta}}=-16\pi G{\sqrt {g\nu }\frac style ^{\nu\frcrt }\fr}\fr}\t }g_{\frcr}\t }\t }\text }$

라이트맨리는^[44] 벨린판테와 시하트의 ^[25][26]비미터 이론을 바탕으로 미터법을 개발했다.그 결과는 BSLL 이론으로 알려져 있습니다.텐서 $필드{\displaystyle {}_{}}$ ${\displaystyle B_{}}$ ${\displaystyle {\mu }_{}{}_{}}$† {$displaystyle B_{\mu \nu$ $B{\displaystyle }$ ${\displaystyle =}$ ${\displaystyle {\mu }^{}}$ ${\displaystyle {\mu }^{}{}^{}}$† {$display$ B$=B_{\mu \nu$}\$eta ^{\mu \nu$ $그리고{\displaystyle }$ 두 $개$의 상수 $a${$displaystyle$ a$,}$ 및$$$$f {\$displaystyle$ f$}$가 주어진 경우 동작은 다음과 같습니다.

${\displaystyle S={1 \over 16\pi G}\int d^{4}xsvrt {-\eta}}(aB^{\mu \nu \alpha})B_{\mu \nu \alpha}+fB^{,\alpha}+S_{m}}$

응력-에너지 텐서는 다음에서 나온다.

${\displaystyle a\Box _{\eta }B^{\mu \nu }+f\eta ^{\mu \nu }\B=-4\pi G{\sqrt {g/\eta }}}, T^{\alpha }\frac {\frac } } } } } 。$

Rastall에서 ^[48]메트릭은 민코프스키 메트릭과 벡터 ^[53]필드의 대수 함수이다.액션:

$\displaystyle S={1 \over 16 \pi G}\int d^{4}x,{\displayrt {-g}}F^{\mu ;\nu}K_{\mu ;\nu }+S_{m}$

어디에

${\displaystyle F\frac{}{}}$(${\displaystyle }$N ) ${\displaystyle =}$ - N ${\displaystyle \frac{2\mathrm{}}{}}$+$$ ( \ $displaystyle$$F$ ( N ) = - { \ $frac${$N$} { $2$+ $N$} 、 N$$${\displaystyle }$ ${\displaystyle =}$ ${\displaystyle {}^{}}$ ${\displaystyle {\mu }_{}}$ K ${\displaystyle \{\backslash {}_{}}$ { \ $displaystyle$ N =$$g$^$ { \ $mu }$K _ { \ $mu$}$K$_ { \ { \ nu $}$ ;

(${\displaystyle {\mathrm{T\mu }}^{}}$ ${\displaystyle \{\backslash {}^{}}$ {\$displaystyle$ T$^{\mu \nu }\;}$ 및$$ $displaystyle K_{\mu$}\;})$$의 $필드{\displaystyle {}^{}}$ 방정식은 Will을 참조하십시오^[10]).

준선형 이론

Whitehead에서는 ^[6]물리적 $메트릭{\displaystyle }$g(\$displaystyle$ g$\;)$는 Minkowski $메트릭{\displaystyle }${\(\displaystyle $\eta \;)$ 및 물질$$ 변수에서 대수적으로 구성되므로 스칼라 필드도 없습니다.구조는 다음과 같습니다.

${\displaystyle g_{\mu \nu }(x^{\alpha })=\eta _{\mu \nu }-2\int _{\Sigma ^{-}}{y_{\mu }^{-}y_{\nu }^{-} \over (w^{-})^{3}}\left[{\sqrt {-g}}\rho u^{\alpha }\,d\Sigma _{\alpha }\right]^{-}}$

where the superscript (−) indicates quantities evaluated along the past ${\displaystyle \eta \;}$ light cone of the field point ${\displaystyle x^{\alpha }\;}$ and

${\displaystyle {\begin{aligned}(y^{\mu })^{-}&=x^{\mu }-(x^{\mu })^{-},\qquad (y^{\mu })^{-}(y_{\mu })^{-}=0,\\[5pt]w^{-}&=(y^{\mu })^{-}(u_{\mu })^{-},\qquad (u_{\mu })={\frac {dx^{\mu }}{d\sigma }},\\[5pt]d\sigma ^{2}&=\eta _{\mu \nu }\,dx^{\mu }\,dx^{\nu }\end{aligned}}}$

Nevertheless, the metric construction (from a non-metric theory) using the "length contraction" ansatz is criticised.^[54]

Deser and Laurent^[33] and Bollini–Giambiagi–Tiomno^[36] are Linear Fixed Gauge theories. Taking an approach from quantum field theory, combine a Minkowski spacetime with the gauge invariant action of a spin-two tensor field (i.e. graviton) ${\displaystyle h_{\mu \nu }\;}$ to define

${\displaystyle g_{\mu \nu }=\eta _{\mu \nu }+h_{\mu \nu }\;}$

The action is:

${\displaystyle S={1 \over 16\pi G}\int d^{4}x{\sqrt {-\eta }}\left[2h_{ \nu }^{\mu \nu }h_{\mu \lambda }^{ \lambda }-2h_{ \nu }^{\mu \nu }h_{\lambda \mu }^{\lambda }+h_{\nu \mu }^{\nu }h_{\lambda }^{\lambda \mu }-h^{\mu \nu \lambda }h_{\mu \nu \lambda }\right]+S_{m}\;}$

The Bianchi identity associated with this partial gauge invariance is wrong. Linear Fixed Gauge theories seek to remedy this by breaking the gauge invariance of the gravitational action through the introduction of auxiliary gravitational fields that couple to ${\displaystyle h_{\mu \nu }\;}$.

A cosmological constant can be introduced into a quasilinear theory by the simple expedient of changing the Minkowski background to a de Sitter or anti-de Sitter spacetime, as suggested by G. Temple in 1923. Temple's suggestions on how to do this were criticized by C. B. Rayner in 1955.^[55]

Tensor theories

Einstein's general relativity is the simplest plausible theory of gravity that can be based on just one symmetric tensor field (the metric tensor). Others include: Starobinsky (R+R^2) gravity, Gauss–Bonnet gravity, f(R) gravity, and Lovelock theory of gravity.

Starobinsky

Starobinsky gravity, proposed by Alexei Starobinsky has the Lagrangian

${\displaystyle {\mathcal {L}}={\sqrt {-g}}\left[R+{\frac {R^{2}}{6M^{2}}}\right]}$

and has been used to explain inflation, in the form of Starobinsky inflation. Here ${\displaystyle M}$ is a constant.

Gauss–Bonnet

Gauss–Bonnet gravity has the action

${\displaystyle {\mathcal {L}}={\sqrt {-g}}\left[R+R^{2}-4R^{\mu \nu }R_{\mu \nu }+R^{\mu \nu \rho \sigma }R_{\mu \nu \rho \sigma }\right].}$

where the coefficients of the extra terms are chosen so that the action reduces to general relativity in 4 spacetime dimensions and the extra terms are only non-trivial when more dimensions are introduced.

Stelle's 4th derivative gravity

Stelle's 4th derivative gravity, which is a generalisation of Gauss–Bonnet gravity, has the action

${\displaystyle {\mathcal {L}}={\sqrt {-g}}\left[R+f_{1}R^{2}+f_{2}R^{\mu \nu }R_{\mu \nu }+f_{3}R^{\mu \nu \rho \sigma }R_{\mu \nu \rho \sigma }\right].}$

f(R)

f(R) gravity has the action

${\displaystyle {\mathcal {L}}={\sqrt {-g}}f(R)}$

and is a family of theories, each defined by a different function of the Ricci scalar. Starobinsky gravity is actually an ${\displaystyle f(R)}$ theory.

Infinite derivative gravity

Infinite derivative gravity is a covariant theory of gravity, quadratic in curvature, torsion free and parity invariant,^[56]

${\displaystyle {\mathcal {L}}={\sqrt {-g}}\left[M_{p}^{2}R+Rf_{1}\left({\frac {\Box }{M_{s}^{2}}}\right)R+R^{\mu \nu }f_{2}\left({\frac {\Box }{M_{s}^{2}}}\right)R_{\mu \nu }+R^{\mu \nu \rho \sigma }f_{3}\left({\frac {\Box }{M_{s}^{2}}}\right)R_{\mu \nu \rho \sigma }\right].}$

and

${\displaystyle 2f_{1}\left({\frac {\Box }{M_{s}^{2}}}\right)+f_{2}\left({\frac {\Box }{M_{s}^{2}}}\right)+2f_{3}\left({\frac {\Box }{M_{s}^{2}}}\right)=0,}$

in order to make sure that only massless spin −2 and spin −0 components propagate in the graviton propagator around Minkowski background. The action becomes non-local beyond the scale ${\displaystyle M_{s}}$, and recovers to general relativity in the infrared, for energies below the non-local scale ${\displaystyle M_{s}}$. In the ultraviolet regime, at distances and time scales below non-local scale, ${\displaystyle M_{s}^{-1}}$, the gravitational interaction weakens enough to resolve point-like singularity, which means Schwarzschild's singularity can be potentially resolved in infinite derivative theories of gravity.

Lovelock

Lovelock gravity has the action

${\displaystyle {\mathcal {L}}={\sqrt {-g}}\ (\alpha _{0}+\alpha _{1}R+\alpha _{2}\left(R^{2}+R_{\alpha \beta \mu \nu }R^{\alpha \beta \mu \nu }-4R_{\mu \nu }R^{\mu \nu }\right)+\alpha _{3}{\mathcal {O}}(R^{3})),}$

and can be thought of as a generalisation of general relativity.

Scalar–tensor theories

These all contain at least one free parameter, as opposed to general relativity which has no free parameters.

Although not normally considered a Scalar–Tensor theory of gravity, the 5 by 5 metric of Kaluza–Klein reduces to a 4 by 4 metric and a single scalar. So if the 5th element is treated as a scalar gravitational field instead of an electromagnetic field then Kaluza–Klein can be considered the progenitor of Scalar–Tensor theories of gravity. This was recognised by Thiry.^[19]

Scalar–Tensor theories include Thiry,^[19] Jordan,^[23] Brans and Dicke,^[8] Bergman,^[35] Nordtveldt (1970), Wagoner,^[38] Bekenstein^[46] and Barker.^[47]

The action ${\displaystyle S\;}$ is based on the integral of the Lagrangian ${\displaystyle L_{\varphi }\;}$.

${\displaystyle S={1 \over 16\pi G}\int d^{4}x{\sqrt {-g}}L_{\varphi }+S_{m}\;}$

${\displaystyle L_{\varphi }=\varphi R-{\omega (\varphi ) \over \varphi }g^{\mu \nu }\,\partial _{\mu }\varphi \,\partial _{\nu }\varphi +2\varphi \lambda (\varphi )\;}$

${\displaystyle S_{m}=\int d^{4}x\,{\sqrt {g}}\,G_{N}L_{m}\;}$

${\displaystyle T^{\mu \nu }\ {\stackrel {\mathrm {def} }{=}}\ {2 \over {\sqrt {g}}}{\delta S_{m} \over \delta g_{\mu \nu }}}$

where ${\displaystyle \omega (\varphi )\;}$ is a different dimensionless function for each different scalar–tensor theory. The function ${\displaystyle \lambda (\varphi )\;}$ plays the same role as the cosmological constant in general relativity. ${\displaystyle G_{N}\;}$ is a dimensionless normalization constant that fixes the present-day value of ${\displaystyle G\;}$. An arbitrary potential can be added for the scalar.

The full version is retained in Bergman^[35] and Wagoner.^[38] Special cases are:

Nordtvedt,^[37] ${\displaystyle \lambda =0\;}$

Since ${\displaystyle \lambda }$ was thought to be zero at the time anyway, this would not have been considered a significant difference. The role of the cosmological constant in more modern work is discussed under Cosmological constant.

Brans–Dicke,^[8] ${\displaystyle \omega \;}$ is constant

Bekenstein^[46] variable mass theory Starting with parameters ${\displaystyle r\;}$ and ${\displaystyle q\;}$, found from a cosmological solution, ${\displaystyle \varphi =[1-qf(\varphi )]f(\varphi )^{-r}\;}$ determines function ${\displaystyle f\;}$ then

${\displaystyle \omega (\varphi )=-\textstyle {\frac {3}{2}}-\textstyle {\frac {1}{4}}f(\varphi )[(1-6q)qf(\varphi )-1][r+(1-r)qf(\varphi )]^{-2}\;}$

Barker^[47] constant G theory

${\displaystyle \omega (\varphi )={\frac {4-3\varphi }{2\varphi -2}}}$

Adjustment of ${\displaystyle \omega (\varphi )\;}$ allows Scalar Tensor Theories to tend to general relativity in the limit of ${\displaystyle \omega \rightarrow \infty \;}$ in the current epoch. However, there could be significant differences from general relativity in the early universe.

So long as general relativity is confirmed by experiment, general Scalar–Tensor theories (including Brans–Dicke^[8]) can never be ruled out entirely, but as experiments continue to confirm general relativity more precisely and the parameters have to be fine-tuned so that the predictions more closely match those of general relativity.

The above examples are particular cases of Horndeski's theory,^[57][58] the most general Lagrangian constructed out of the metric tensor and a scalar field leading to second order equations of motion in 4-dimensional space. Viable theories beyond Horndeski (with higher order equations of motion) have been shown to exist.^[59][60][61]

Vector–tensor theories

Before we start, Will (2001) has said: "Many alternative metric theories developed during the 1970s and 1980s could be viewed as "straw-man" theories, invented to prove that such theories exist or to illustrate particular properties. Few of these could be regarded as well-motivated theories from the point of view, say, of field theory or particle physics. Examples are the vector–tensor theories studied by Will, Nordtvedt and Hellings."

Hellings and Nordtvedt^[43] and Will and Nordtvedt^[42] are both vector–tensor theories. In addition to the metric tensor there is a timelike vector field ${\displaystyle K_{\mu }.}$ The gravitational action is:

${\displaystyle S={\frac {1}{16\pi G}}\int d^{4}x{\sqrt {-g}}\left[R+\omega K_{\mu }K^{\mu }R+\eta K^{\mu }K^{\nu }R_{\mu \nu }-\epsilon F_{\mu \nu }F^{\mu \nu }+\tau K_{\mu ;\nu }K^{\mu ;\nu }\right]+S_{m}}$

where ${\displaystyle \omega ,\eta ,\epsilon ,\tau }$ are constants and

${\displaystyle F_{\mu \nu }=K_{\nu ;\mu }-K_{\mu ;\nu }.}$ (See Will^[10] for the field equations for ${\displaystyle T^{\mu \nu }}$ and ${\displaystyle K_{\mu }.}$)

Will and Nordtvedt^[42] is a special case where

${\displaystyle \omega =\eta =\epsilon =0;\quad \tau =1}$

Hellings and Nordtvedt^[43] is a special case where

${\displaystyle \tau =0;\quad \epsilon =1;\quad \eta =-2\omega }$

These vector–tensor theories are semi-conservative, which means that they satisfy the laws of conservation of momentum and angular momentum but can have preferred frame effects. When ${\displaystyle \omega =\eta =\epsilon =\tau =0}$ they reduce to general relativity so, so long as general relativity is confirmed by experiment, general vector–tensor theories can never be ruled out.

Other metric theories

Others metric theories have been proposed; that of Bekenstein^[62] is discussed under Modern Theories.

Non-metric theories

Cartan's theory is particularly interesting both because it is a non-metric theory and because it is so old. The status of Cartan's theory is uncertain. Will^[10] claims that all non-metric theories are eliminated by Einstein's Equivalence Principle. Will (2001) tempers that by explaining experimental criteria for testing non-metric theories against Einstein's Equivalence Principle. Misner et al.^[51] claims that Cartan's theory is the only non-metric theory to survive all experimental tests up to that date and Turyshev^[63] lists Cartan's theory among the few that have survived all experimental tests up to that date. The following is a quick sketch of Cartan's theory as restated by Trautman.^[64]

Cartan^[14][15] suggested a simple generalization of Einstein's theory of gravitation. He proposed a model of space time with a metric tensor and a linear "connection" compatible with the metric but not necessarily symmetric. The torsion tensor of the connection is related to the density of intrinsic angular momentum. Independently of Cartan, similar ideas were put forward by Sciama, by Kibble in the years 1958 to 1966, culminating in a 1976 review by Hehl et al.

The original description is in terms of differential forms, but for the present article that is replaced by the more familiar language of tensors (risking loss of accuracy). As in general relativity, the Lagrangian is made up of a massless and a mass part. The Lagrangian for the massless part is:

${\displaystyle {\begin{aligned}L&={1 \over 32\pi G}\Omega _{\nu }^{\mu }g^{\nu \xi }x^{\eta }x^{\zeta }\varepsilon _{\xi \mu \eta \zeta }\\[5pt]\Omega _{\nu }^{\mu }&=d\omega _{\nu }^{\mu }+\omega _{\xi }^{\eta }\\[5pt]\nabla x^{\mu }&=-\omega _{\nu }^{\mu }x^{\nu }\end{aligned}}}$

The ${\displaystyle \omega _{\nu }^{\mu }\;}$ is the linear connection. ${\displaystyle \varepsilon _{\xi \mu \eta \zeta }\;}$ is the completely antisymmetric pseudo-tensor (Levi-Civita symbol) with ${\displaystyle \varepsilon _{0123}={\sqrt {-g}}\;}$, and ${\displaystyle g^{\nu \xi }\,}$ is the metric tensor as usual. By assuming that the linear connection is metric, it is possible to remove the unwanted freedom inherent in the non-metric theory. The stress–energy tensor is calculated from:

${\displaystyle T^{\mu \nu }={1 \over 16\pi G}(g^{\mu \nu }\eta _{\eta }^{\xi }-g^{\xi \mu }\eta _{\eta }^{\nu }-g^{\xi \nu }\eta _{\eta }^{\mu })\Omega _{\xi }^{\eta }\;}$

The space curvature is not Riemannian, but on a Riemannian space-time the Lagrangian would reduce to the Lagrangian of general relativity.

Some equations of the non-metric theory of Belinfante and Swihart^[25][26] have already been discussed in the section on bimetric theories.

A distinctively non-metric theory is given by gauge theory gravity, which replaces the metric in its field equations with a pair of gauge fields in flat spacetime. On the one hand, the theory is quite conservative because it is substantially equivalent to Einstein–Cartan theory (or general relativity in the limit of vanishing spin), differing mostly in the nature of its global solutions. On the other hand, it is radical because it replaces differential geometry with geometric algebra.

Modern theories 1980s to present

This section includes alternatives to general relativity published after the observations of galaxy rotation that led to the hypothesis of "dark matter". There is no known reliable list of comparison of these theories. Those considered here include: Bekenstein,^[62] Moffat,^[65] Moffat,^[66] Moffat.^[67][68] These theories are presented with a cosmological constant or added scalar or vector potential.

Motivations

Motivations for the more recent alternatives to general relativity are almost all cosmological, associated with or replacing such constructs as "inflation", "dark matter" and "dark energy". The basic idea is that gravity agrees with general relativity at the present epoch but may have been quite different in the early universe.

In the 1980s, there was a slowly dawning realisation in the physics world that there were several problems inherent in the then-current big-bang scenario, including the horizon problem and the observation that at early times when quarks were first forming there was not enough space on the universe to contain even one quark. Inflation theory was developed to overcome these difficulties. Another alternative was constructing an alternative to general relativity in which the speed of light was higher in the early universe. The discovery of unexpected rotation curves for galaxies took everyone by surprise. Could there be more mass in the universe than we are aware of, or is the theory of gravity itself wrong? The consensus now is that the missing mass is "cold dark matter", but that consensus was only reached after trying alternatives to general relativity, and some physicists still believe that alternative models of gravity may hold the answer.

In the 1990s, supernova surveys discovered the accelerated expansion of the universe, now usually attributed to dark energy. This led to the rapid reinstatement of Einstein's cosmological constant, and quintessence arrived as an alternative to the cosmological constant. At least one new alternative to general relativity attempted to explain the supernova surveys' results in a completely different way. The measurement of the speed of gravity with the gravitational wave event GW170817 ruled out many alternative theories of gravity as explanations for the accelerated expansion.^[69][70][71] Another observation that sparked recent interest in alternatives to General Relativity is the Pioneer anomaly. It was quickly discovered that alternatives to general relativity could explain this anomaly. This is now believed to be accounted for by non-uniform thermal radiation.

Cosmological constant and quintessence

The cosmological constant ${\displaystyle \Lambda \;}$ is a very old idea, going back to Einstein in 1917.^[4] The success of the Friedmann model of the universe in which ${\displaystyle \Lambda =0\;}$ led to the general acceptance that it is zero, but the use of a non-zero value came back with a vengeance when data from supernovae indicated that the expansion of the universe is accelerating

First, let's see how it influences the equations of Newtonian gravity and General Relativity. In Newtonian gravity, the addition of the cosmological constant changes the Newton–Poisson equation from:

${\displaystyle \nabla ^{2}\varphi =4\pi \rho \ G;}$

to

${\displaystyle \nabla ^{2}\varphi +{\frac {1}{2}}\Lambda c^{2}=4\pi \rho \ G;}$

In general relativity, it changes the Einstein–Hilbert action from

${\displaystyle S={1 \over 16\pi G}\int R{\sqrt {-g}}\,d^{4}x\,+S_{m}\;}$

to

${\displaystyle S={1 \over 16\pi G}\int (R-2\Lambda ){\sqrt {-g}}\,d^{4}x\,+S_{m}\;}$

which changes the field equation

${\displaystyle T^{\mu \nu }={1 \over 8\pi G}\left(R^{\mu \nu }-{\frac {1}{2}}g^{\mu \nu }R\right)\;}$

to

${\displaystyle T^{\mu \nu }={1 \over 8\pi G}\left(R^{\mu \nu }-{\frac {1}{2}}g^{\mu \nu }R+g^{\mu \nu }\Lambda \right)\;}$

In alternative theories of gravity, a cosmological constant can be added to the action in exactly the same way.

The cosmological constant is not the only way to get an accelerated expansion of the universe in alternatives to general relativity. We've already seen how the scalar potential ${\displaystyle \lambda (\varphi )\;}$ can be added to scalar tensor theories. This can also be done in every alternative the general relativity that contains a scalar field ${\displaystyle \varphi \;}$ by adding the term ${\displaystyle \lambda (\varphi )\;}$ inside the Lagrangian for the gravitational part of the action, the ${\displaystyle L_{\varphi }\;}$ part of

${\displaystyle S={1 \over 16\pi G}\int d^{4}x\,{\sqrt {-g}}\,L_{\varphi }+S_{m}\;}$

Because ${\displaystyle \lambda (\varphi )\;}$ is an arbitrary function of the scalar field, it can be set to give an acceleration that is large in the early universe and small at the present epoch. This is known as quintessence.

A similar method can be used in alternatives to general relativity that use vector fields, including Rastall^[48] and vector–tensor theories. A term proportional to

${\displaystyle K^{\mu }K^{\nu }g_{\mu \nu }\;}$

is added to the Lagrangian for the gravitational part of the action.

Farnes' theories

In December 2018, the astrophysicist Jamie Farnes from the University of Oxford proposed a dark fluid theory, related to notions of gravitationally repulsive negative masses that were presented earlier by Albert Einstein. The theory may help to better understand the considerable amounts of unknown dark matter and dark energy in the universe.^[72]

The theory relies on the concept of negative mass and reintroduces Fred Hoyle's creation tensor in order to allow matter creation for only negative mass particles. In this way, the negative mass particles surround galaxies and apply a pressure onto them, thereby resembling dark matter. As these hypothesised particles mutually repel one another, they push apart the Universe, thereby resembling dark energy. The creation of matter allows the density of the exotic negative mass particles to remain constant as a function of time, and so appears like a cosmological constant. Einstein's field equations are modified to:

${\displaystyle R_{\mu \nu }-{\frac {1}{2}}Rg_{\mu \nu }={\frac {8\pi G}{c^{4}}}\left(T_{\mu \nu }^{+}+T_{\mu \nu }^{-}+C_{\mu \nu }\right)}$

According to Occam's razor, Farnes' theory is a simpler alternative to the conventional LambdaCDM model, as both dark energy and dark matter (two hypotheses) are solved using a single negative mass fluid (one hypothesis). The theory will be directly testable using the world's largest radio telescope, the Square Kilometre Array which should come online in 2022.^[73]

Relativistic MOND

The original theory of MOND by Milgrom was developed in 1983 as an alternative to "dark matter". Departures from Newton's law of gravitation are governed by an acceleration scale, not a distance scale. MOND successfully explains the Tully–Fisher observation that the luminosity of a galaxy should scale as the fourth power of the rotation speed. It also explains why the rotation discrepancy in dwarf galaxies is particularly large.

There were several problems with MOND in the beginning.

1. It did not include relativistic effects
2. It violated the conservation of energy, momentum and angular momentum
3. It was inconsistent in that it gives different galactic orbits for gas and for stars
4. It did not state how to calculate gravitational lensing from galaxy clusters.

By 1984, problems 2 and 3 had been solved by introducing a Lagrangian (AQUAL). A relativistic version of this based on scalar–tensor theory was rejected because it allowed waves in the scalar field to propagate faster than light. The Lagrangian of the non-relativistic form is:

${\displaystyle L=-{a_{0}^{2} \over 8\pi G}f\left\lbrack {\frac { \nabla \varphi ^{2}}{a_{0}^{2}}}\right\rbrack -\rho \varphi }$

The relativistic version of this has:

${\displaystyle L=-{a_{0}^{2} \over 8\pi G}{\tilde {f}}\left(\ell _{0}^{2}g^{\mu \nu }\,\partial _{\mu }\varphi \,\partial _{\nu }\varphi \right)}$

with a nonstandard mass action. Here ${\displaystyle f}$ and ${\displaystyle {\tilde {f}}}$ are arbitrary functions selected to give Newtonian and MOND behaviour in the correct limits, and ${\displaystyle l_{0}=c^{2}/a_{0}\;}$ is the MOND length scale. By 1988, a second scalar field (PCC) fixed problems with the earlier scalar–tensor version but is in conflict with the perihelion precession of Mercury and gravitational lensing by galaxies and clusters. By 1997, MOND had been successfully incorporated in a stratified relativistic theory [Sanders], but as this is a preferred frame theory it has problems of its own. Bekenstein^[62] introduced a tensor–vector–scalar model (TeVeS). This has two scalar fields ${\displaystyle \varphi }$ and ${\displaystyle \sigma \;}$ and vector field ${\displaystyle U_{\alpha }}$. The action is split into parts for gravity, scalars, vector and mass.

${\displaystyle S=S_{g}+S_{s}+S_{v}+S_{m}}$

The gravity part is the same as in general relativity.

${\displaystyle {\begin{aligned}S_{s}&=-{\frac {1}{2}}\int \left[\sigma ^{2}h^{\alpha \beta }\varphi _{,\alpha }\varphi _{,\beta }+{\frac {1}{2}}G\ell _{0}^{-2}\sigma ^{4}F(kG\sigma ^{2})\right]{\sqrt {-g}}\,d^{4}x\\[5pt]S_{v}&=-{\frac {K}{32\pi G}}\int \left[g^{\alpha \beta }g^{\mu \nu }U_{[\alpha ,\mu ]}U_{[\beta ,\nu ]}-{\frac {2\lambda }{K}}\left(g^{\mu \nu }U_{\mu }U_{\nu }+1\right)\right]{\sqrt {-g}}\,d^{4}x\\[5pt]S_{m}&=\int L\left({\tilde {g}}_{\mu \nu },f^{\alpha },f_{ \mu }^{\alpha },\ldots \right){\sqrt {-g}}\,d^{4}x\end{aligned}}}$

where

${\displaystyle h^{\alpha \beta }=g^{\alpha \beta }-U^{\alpha }U^{\beta }}$

${\displaystyle {\tilde {g}}^{\alpha \beta }=e^{2\varphi }g^{\alpha \beta }+2U^{\alpha }U^{\beta }\sinh(2\varphi )}$

${\displaystyle k,K}$ are constants, square brackets in indices ${\displaystyle U_{[\alpha ,\mu ]}}$ represent anti-symmetrization, ${\displaystyle \lambda }$ is a Lagrange multiplier (calculated elsewhere), and L is a Lagrangian translated from flat spacetime onto the metric ${\displaystyle {\tilde {g}}^{\alpha \beta }}$. Note that G need not equal the observed gravitational constant ${\displaystyle G_{Newton}}$. F is an arbitrary function, and

${\displaystyle F(\mu )={\frac {3}{4}}{\mu ^{2}(\mu -2)^{2} \over 1-\mu }}$

is given as an example with the right asymptotic behaviour; note how it becomes undefined when ${\displaystyle \mu =1}$

The Parametric post-Newtonian parameters of this theory are calculated in,^[74] which shows that all its parameters are equal to general relativity's, except for

${\displaystyle {\begin{aligned}\alpha _{1}&={\frac {4G}{K}}\left((2K-1)e^{-4\varphi _{0}}-e^{4\varphi _{0}}+8\right)-8\\[5pt]\alpha _{2}&={\frac {6G}{2-K}}-{\frac {2G(K+4)e^{4\varphi _{0}}}{(2-K)^{2}}}-1\end{aligned}}}$

both of which expressed in geometric units where ${\displaystyle c=G_{Newtonian}=1}$; so

${\displaystyle G^{-1}={\frac {2}{2-K}}+{\frac {k}{4\pi }}.}$

Moffat's theories

J. W. Moffat^[65] developed a non-symmetric gravitation theory. This is not a metric theory. It was first claimed that it does not contain a black hole horizon, but Burko and Ori^[75] have found that nonsymmetric gravitational theory can contain black holes. Later, Moffat claimed that it has also been applied to explain rotation curves of galaxies without invoking "dark matter". Damour, Deser & MaCarthy^[76] have criticised nonsymmetric gravitational theory, saying that it has unacceptable asymptotic behaviour.

The mathematics is not difficult but is intertwined so the following is only a brief sketch. Starting with a non-symmetric tensor ${\displaystyle g_{\mu \nu }\;}$, the Lagrangian density is split into

${\displaystyle L=L_{R}+L_{M}\;}$

where ${\displaystyle L_{M}\;}$ is the same as for matter in general relativity.

${\displaystyle L_{R}={\sqrt {-g}}\left[R(W)-2\lambda -{\frac {1}{4}}\mu ^{2}g^{\mu \nu }g_{[\mu \nu ]}\right]-{\frac {1}{6}}g^{\mu \nu }W_{\mu }W_{\nu }\;}$

where ${\displaystyle R(W)\;}$ is a curvature term analogous to but not equal to the Ricci curvature in general relativity, ${\displaystyle \lambda \;}$ and ${\displaystyle \mu ^{2}\;}$ are cosmological constants, ${\displaystyle g_{[\nu \mu ]}\;}$ is the antisymmetric part of ${\displaystyle g_{\nu \mu }\;}$. ${\displaystyle W_{\mu }\;}$ is a connection, and is a bit difficult to explain because it's defined recursively. However, ${\displaystyle W_{\mu }\approx -2g_{[\mu \nu ]}^{,\nu }\;}$

Haugan and Kauffmann^[77] used polarization measurements of the light emitted by galaxies to impose sharp constraints on the magnitude of some of nonsymmetric gravitational theory's parameters. They also used Hughes-Drever experiments to constrain the remaining degrees of freedom. Their constraint is eight orders of magnitude sharper than previous estimates.

Moffat's^[67] metric-skew-tensor-gravity (MSTG) theory is able to predict rotation curves for galaxies without either dark matter or MOND, and claims that it can also explain gravitational lensing of galaxy clusters without dark matter. It has variable ${\displaystyle G\;}$, increasing to a final constant value about a million years after the big bang.

The theory seems to contain an asymmetric tensor ${\displaystyle A_{\mu \nu }\;}$ field and a source current ${\displaystyle J_{\mu }\;}$ vector. The action is split into:

${\displaystyle S=S_{G}+S_{F}+S_{FM}+S_{M}\;}$

Both the gravity and mass terms match those of general relativity with cosmological constant. The skew field action and the skew field matter coupling are:

${\displaystyle S_{F}=\int d^{4}x\,{\sqrt {-g}}\left({\frac {1}{12}}F_{\mu \nu \rho }F^{\mu \nu \rho }-{\frac {1}{4}}\mu ^{2}A_{\mu \nu }A^{\mu \nu }\right)\;}$

${\displaystyle S_{FM}=\int d^{4}x\,\epsilon ^{\alpha \beta \mu \nu }A_{\alpha \beta }\partial _{\mu }J_{\nu }\;}$

where

${\displaystyle F_{\mu \nu \rho }=\partial _{\mu }A_{\nu \rho }+\partial _{\rho }A_{\mu \nu }}$

and ${\displaystyle \epsilon ^{\alpha \beta \mu \nu }\;}$ is the Levi-Civita symbol. The skew field coupling is a Pauli coupling and is gauge invariant for any source current. The source current looks like a matter fermion field associated with baryon and lepton number.

Scalar–tensor–vector gravity

Moffat's Scalar–tensor–vector gravity^[68] contains a tensor, vector and three scalar fields. But the equations are quite straightforward. The action is split into: ${\displaystyle S=S_{G}+S_{K}+S_{S}+S_{M}}$ with terms for gravity, vector field ${\displaystyle K_{\mu },}$ scalar fields ${\displaystyle G,\omega ,\mu }$ and mass. ${\displaystyle S_{G}}$ is the standard gravity term with the exception that ${\displaystyle G}$ is moved inside the integral.

${\displaystyle S_{K}=-\int d^{4}x\,{\sqrt {-g}}\omega \left({\frac {1}{4}}B_{\mu \nu }B^{\mu \nu }+V(K)\right),\qquad {\text{where }}\quad B_{\mu \nu }=\partial _{\mu }K_{\nu }-\partial _{\nu }K_{\mu }.}$

${\displaystyle S_{S}=-\int d^{4}x\,{\sqrt {-g}}{\frac {1}{G^{3}}}\left({\frac {1}{2}}g^{\mu \nu }\,\nabla _{\mu }G\,\nabla _{\nu }G-V(G)\right)+{\frac {1}{G}}\left({\frac {1}{2}}g^{\mu \nu }\,\nabla _{\mu }\omega \,\nabla _{\nu }\omega -V(\omega )\right)+{1 \over \mu ^{2}G}\left({\frac {1}{2}}g^{\mu \nu }\,\nabla _{\mu }\mu \,\nabla _{\nu }\mu -V(\mu )\right).}$

The potential function for the vector field is chosen to be:

${\displaystyle V(K)=-{\frac {1}{2}}\mu ^{2}\varphi ^{\mu }\varphi _{\mu }-{\frac {1}{4}}g\left(\varphi ^{\mu }\varphi _{\mu }\right)^{2}}$

where ${\displaystyle g}$ is a coupling constant. The functions assumed for the scalar potentials are not stated.

Infinite derivative gravity

In order to remove ghosts in the modified propagator, as well as to obtain asymptotic freedom, Biswas, Mazumdar and Siegel (2005) considered a string-inspired infinite set of higher derivative terms

${\displaystyle S=\int \mathrm {d} ^{4}x{\sqrt {-g}}\left({\frac {R}{2}}+RF(\Box )R\right)}$

where ${\displaystyle F(\Box )}$ is the exponential of an entire function of the D'Alembertian operator.^[78][79] This avoids a black hole singularity near the origin, while recovering the 1/r fall of the general relativity potential at large distances.^[80] Lousto and Mazzitelli (1997) found an exact solution to this theories representing a gravitational shock-wave.^[81]

Testing of alternatives to general relativity

Any putative alternative to general relativity would need to meet a variety of tests for it to become accepted. For in-depth coverage of these tests, see Misner et al.^[51] Ch.39, Will ^[10] Table 2.1, and Ni.^[11] Most such tests can be categorized as in the following subsections.

Self-consistency

Self-consistency among non-metric theories includes eliminating theories allowing tachyons, ghost poles and higher order poles, and those that have problems with behaviour at infinity. Among metric theories, self-consistency is best illustrated by describing several theories that fail this test. The classic example is the spin-two field theory of Fierz and Pauli;^[16] the field equations imply that gravitating bodies move in straight lines, whereas the equations of motion insist that gravity deflects bodies away from straight line motion. Yilmaz (1971)^[28] contains a tensor gravitational field used to construct a metric; it is mathematically inconsistent because the functional dependence of the metric on the tensor field is not well defined.

Completeness

To be complete, a theory of gravity must be capable of analysing the outcome of every experiment of interest. It must therefore mesh with electromagnetism and all other physics. For instance, any theory that cannot predict from first principles the movement of planets or the behaviour of atomic clocks is incomplete.

Many early theories are incomplete in that it is unclear whether the density ${\displaystyle \rho }$ used by the theory should be calculated from the stress–energy tensor ${\displaystyle T}$ as ${\displaystyle \rho =T_{\mu \nu }u^{\mu }u^{\nu }}$ or as ${\displaystyle \rho =T_{\mu \nu }\delta ^{\mu \nu }}$, where ${\displaystyle u}$ is the four-velocity, and ${\displaystyle \delta }$ is the Kronecker delta. The theories of Thirry (1948) and Jordan^[23] are incomplete unless Jordan's parameter ${\displaystyle \eta \;}$ is set to -1, in which case they match the theory of Brans–Dicke^[8] and so are worthy of further consideration. Milne^[18] is incomplete because it makes no gravitational red-shift prediction. The theories of Whitrow and Morduch,^[29][30] Kustaanheimo^[31] and Kustaanheimo and Nuotio^[32] are either incomplete or inconsistent. The incorporation of Maxwell's equations is incomplete unless it is assumed that they are imposed on the flat background space-time, and when that is done they are inconsistent, because they predict zero gravitational redshift when the wave version of light (Maxwell theory) is used, and nonzero redshift when the particle version (photon) is used. Another more obvious example is Newtonian gravity with Maxwell's equations; light as photons is deflected by gravitational fields (by half that of general relativity) but light as waves is not.

Classical tests

There are three "classical" tests (dating back to the 1910s or earlier) of the ability of gravity theories to handle relativistic effects; they are gravitational redshift, gravitational lensing (generally tested around the Sun), and anomalous perihelion advance of the planets. Each theory should reproduce the observed results in these areas, which have to date always aligned with the predictions of general relativity. In 1964, Irwin I. Shapiro found a fourth test, called the Shapiro delay. It is usually regarded as a "classical" test as well.

Agreement with Newtonian mechanics and special relativity

As an example of disagreement with Newtonian experiments, Birkhoff^[17] theory predicts relativistic effects fairly reliably but demands that sound waves travel at the speed of light. This was the consequence of an assumption made to simplify handling the collision of masses.^{[citation needed]}

The Einstein equivalence principle

Einstein's Equivalence Principle has three components. The first is the uniqueness of free fall, also known as the Weak Equivalence Principle. This is satisfied if inertial mass is equal to gravitational mass. η is a parameter used to test the maximum allowable violation of the Weak Equivalence Principle. The first tests of the Weak Equivalence Principle were done by Eötvös before 1900 and limited η to less than 5×10⁻⁹. Modern tests have reduced that to less than 5×10⁻¹³. The second is Lorentz invariance. In the absence of gravitational effects the speed of light is constant. The test parameter for this is δ. The first tests of Lorentz invariance were done by Michelson and Morley before 1890 and limited δ to less than 5×10⁻³. Modern tests have reduced this to less than 1×10⁻²¹. The third is local position invariance, which includes spatial and temporal invariance. The outcome of any local non-gravitational experiment is independent of where and when it is performed. Spatial local position invariance is tested using gravitational redshift measurements. The test parameter for this is α. Upper limits on this found by Pound and Rebka in 1960 limited α to less than 0.1. Modern tests have reduced this to less than 1×10⁻⁴.

Schiff's conjecture states that any complete, self-consistent theory of gravity that embodies the Weak Equivalence Principle necessarily embodies Einstein's Equivalence Principle. This is likely to be true if the theory has full energy conservation. Metric theories satisfy the Einstein Equivalence Principle. Extremely few non-metric theories satisfy this. For example, the non-metric theory of Belinfante & Swihart^[25][26] is eliminated by the THεμ formalism for testing Einstein's Equivalence Principle. Gauge theory gravity is a notable exception, where the strong equivalence principle is essentially the minimal coupling of the gauge covariant derivative.

Parametric post-Newtonian formalism

See also Tests of general relativity, Misner et al.^[51] and Will^[10] for more information.

Work on developing a standardized rather than ad hoc set of tests for evaluating alternative gravitation models began with Eddington in 1922 and resulted in a standard set of Parametric post-Newtonian numbers in Nordtvedt and Will^[82] and Will and Nordtvedt.^[42] Each parameter measures a different aspect of how much a theory departs from Newtonian gravity. Because we are talking about deviation from Newtonian theory here, these only measure weak-field effects. The effects of strong gravitational fields are examined later.

These ten are: ${\displaystyle \gamma ,\beta ,\eta ,\alpha _{1},\alpha _{2},\alpha _{3},\zeta _{1},\zeta _{2},\zeta _{3},\zeta _{4}.}$

• ${\displaystyle \gamma }$ is a measure of space curvature, being zero for Newtonian gravity and one for general relativity.
• ${\displaystyle \beta }$ is a measure of nonlinearity in the addition of gravitational fields, one for general relativity.
• ${\displaystyle \eta }$ is a check for preferred location effects.
• ${\displaystyle \alpha _{1},\alpha _{2},\alpha _{3}}$ measure the extent and nature of "preferred-frame effects". Any theory of gravity in which at least one of the three is nonzero is called a preferred-frame theory.
• ${\displaystyle \zeta _{1},\zeta _{2},\zeta _{3},\zeta _{4},\alpha _{3}}$ measure the extent and nature of breakdowns in global conservation laws. A theory of gravity possesses 4 conservation laws for energy-momentum and 6 for angular momentum only if all five are zero.

Strong gravity and gravitational waves

Parametric post-Newtonian is only a measure of weak field effects. Strong gravity effects can be seen in compact objects such as white dwarfs, neutron stars, and black holes. Experimental tests such as the stability of white dwarfs, spin-down rate of pulsars, orbits of binary pulsars and the existence of a black hole horizon can be used as tests of alternative to general relativity. General relativity predicts that gravitational waves travel at the speed of light. Many alternatives to general relativity say that gravitational waves travel faster than light, possibly breaking causality. After the multi-messaging detection of the GW170817 coalescence of neutron stars, where light and gravitational waves were measured to travel at the same speed with an error of 1/10¹⁵, many of those modified theory of gravity were excluded.

Cosmological tests

Many of these have been developed recently. For those theories that aim to replace dark matter, the galaxy rotation curve, the Tully–Fisher relation, the faster rotation rate of dwarf galaxies, and the gravitational lensing due to galactic clusters act as constraints. For those theories that aim to replace inflation, the size of ripples in the spectrum of the cosmic microwave background radiation is the strictest test. For those theories that incorporate or aim to replace dark energy, the supernova brightness results and the age of the universe can be used as tests. Another test is the flatness of the universe. With general relativity, the combination of baryonic matter, dark matter and dark energy add up to make the universe exactly flat. As the accuracy of experimental tests improve, alternatives to general relativity that aim to replace dark matter or dark energy will have to explain why.

Results of testing theories

Parametric post-Newtonian parameters for a range of theories

(See Will^[10] and Ni^[11] for more details. Misner et al.^[51] gives a table for translating parameters from the notation of Ni to that of Will)

General Relativity is now more than 100 years old, during which one alternative theory of gravity after another has failed to agree with ever more accurate observations. One illustrative example is Parameterized post-Newtonian formalism. The following table lists Parametric post-Newtonian values for a large number of theories. If the value in a cell matches that in the column heading then the full formula is too complicated to include here.

	${\displaystyle \gamma }$	${\displaystyle \beta }$	${\displaystyle \xi }$	${\displaystyle \alpha _{1}}$	${\displaystyle \alpha _{2}}$	${\displaystyle \alpha _{3}}$	${\displaystyle \zeta _{1}}$	${\displaystyle \zeta _{2}}$	${\displaystyle \zeta _{3}}$	${\displaystyle \zeta _{4}}$
Einstein general relativity[3]	1	1	0	0	0	0	0	0	0	0
Scalar–tensor theories
Bergmann,[35] Wagoner[38]	${\displaystyle \textstyle {\frac {1+\omega }{2+\omega }}}$	${\displaystyle \beta }$	0	0	0	0	0	0	0	0
Nordtvedt,[37] Bekenstein[46]	${\displaystyle \textstyle {\frac {1+\omega }{2+\omega }}}$	${\displaystyle \beta }$	0	0	0	0	0	0	0	0
Brans–Dicke[8]	${\displaystyle \textstyle {\frac {1+\omega }{2+\omega }}}$	1	0	0	0	0	0	0	0	0
Vector–tensor theories
Hellings-Nordtvedt[43]	${\displaystyle \gamma }$	${\displaystyle \beta }$	0	${\displaystyle \alpha _{1}}$	${\displaystyle \alpha _{2}}$	0	0	0	0	0
Will-Nordtvedt[42]	1	1	0	0	${\displaystyle \alpha _{2}}$	0	0	0	0	0
Bimetric theories
Rosen[40]	1	1	0	0	${\displaystyle c_{0}/c_{1}-1}$	0	0	0	0	0
Rastall[48]	1	1	0	0	${\displaystyle \alpha _{2}}$	0	0	0	0	0
Lightman–Lee[44]	${\displaystyle \gamma }$	${\displaystyle \beta }$	0	${\displaystyle \alpha _{1}}$	${\displaystyle \alpha _{2}}$	0	0	0	0	0
Stratified theories
Lee–Lightman–Ni[45]	${\displaystyle ac_{0}/c_{1}}$	${\displaystyle \beta }$	${\displaystyle \xi }$	${\displaystyle \alpha _{1}}$	${\displaystyle \alpha _{2}}$	0	0	0	0	0
Ni[41]	${\displaystyle ac_{0}/c_{1}}$	${\displaystyle bc_{0}}$	0	${\displaystyle \alpha _{1}}$	${\displaystyle \alpha _{2}}$	0	0	0	0	0
Scalar field theories
Einstein (1912)[83][84] {Not general relativity}	0	0		-4	0	-2	0	-1	0	0†
Whitrow–Morduch[30]	0	-1		-4	0	0	0	−3	0	0†
Rosen[39]	${\displaystyle \lambda }$	${\displaystyle \textstyle {\frac {3}{4}}+\textstyle {\frac {\lambda }{4}}}$		${\displaystyle -4-4\lambda }$	0	-4	0	-1	0	0
Papapetrou[20][21]	1	1		-8	-4	0	0	2	0	0
Ni[11] (stratified)	1	1		-8	0	0	0	2	0	0
Yilmaz[27] (1962)	1	1		-8	0	-4	0	-2	0	-1†
Page–Tupper[34]	${\displaystyle \gamma }$	${\displaystyle \beta }$		${\displaystyle -4-4\gamma }$	0	${\displaystyle -2-2\gamma }$	0	${\displaystyle \zeta _{2}}$	0	${\displaystyle \zeta _{4}}$
Nordström[49]	${\displaystyle -1}$	${\displaystyle \textstyle {\frac {1}{2}}}$		0	0	0	0	0	0	0†
Nordström,[50] Einstein-Fokker[85]	${\displaystyle -1}$	${\displaystyle \textstyle {\frac {1}{2}}}$		0	0	0	0	0	0	0
Ni[11] (flat)	${\displaystyle -1}$	${\displaystyle 1-q}$		0	0	0	0	${\displaystyle \zeta _{2}}$	0	0†
Whitrow–Morduch[29]	${\displaystyle -1}$	${\displaystyle 1-q}$		0	0	0	0	q	0	0†
Littlewood,[22] Bergman[24]	${\displaystyle -1}$	${\displaystyle \textstyle {\frac {1}{2}}}$		0	0	0	0	-1	0	0†

† The theory is incomplete, and ${\displaystyle \zeta _{4}}$ can take one of two values. The value closest to zero is listed.

All experimental tests agree with general relativity so far, and so Parametric post-Newtonian analysis immediately eliminates all the scalar field theories in the table. A full list of Parametric post-Newtonian parameters is not available for Whitehead,^[6] Deser-Laurent,^[33] Bollini–Giambiagi–Tiomino,^[36] but in these three cases ${\displaystyle \beta =\xi }$,^{[citation needed]} which is in strong conflict with general relativity and experimental results. In particular, these theories predict incorrect amplitudes for the Earth's tides. (A minor modification of Whitehead's theory avoids this problem. However, the modification predicts the Nordtvedt effect, which has been experimentally constrained.)

Theories that fail other tests

The stratified theories of Ni,^[41] Lee Lightman and Ni^[45] are non-starters because they all fail to explain the perihelion advance of Mercury. The bimetric theories of Lightman and Lee,^[44] Rosen,^[40] Rastall^[48] all fail some of the tests associated with strong gravitational fields. The scalar–tensor theories include general relativity as a special case, but only agree with the Parametric post-Newtonian values of general relativity when they are equal to general relativity to within experimental error. As experimental tests get more accurate, the deviation of the scalar–tensor theories from general relativity is being squashed to zero. The same is true of vector–tensor theories, the deviation of the vector–tensor theories from general relativity is being squashed to zero. Further, vector–tensor theories are semi-conservative; they have a nonzero value for ${\displaystyle \alpha _{2}}$ which can have a measurable effect on the Earth's tides. Non-metric theories, such as Belinfante and Swihart,^[25][26] usually fail to agree with experimental tests of Einstein's equivalence principle. And that leaves, as a likely valid alternative to general relativity, nothing except possibly Cartan.^[14] That was the situation until cosmological discoveries pushed the development of modern alternatives.

Footnotes

References