廣義相對論的替代理論
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廣義相對論的替代理論是與愛因斯坦廣義相對論(general relativity, GR)競爭，嘗試要描述引力現象的物理理論。
對於建構一個理想引力理論，至今已有許多不同的嚐試。這些嚐試可以分為下面四個大類：
- 與GR直接競爭的理論，例如嘉當(Cartan)理論、布蘭斯-狄基(Brans-Dicke)理論以及 羅森雙度規(Rosen bimetric)等理論。
- 嚐試建構量子化的引力理論，例如圈量子引力論。
- 嚐試統一引力與其他基本力，例如卡魯扎-克萊因理論。
- 嚐試將所有目標畢其功於一役，例如M理論。
本文談論對象僅包括與GR的直接競爭理論。量子化的引力理論課題，請見量子引力條目。引力與其他基本力的統一理論課題，請見经典統一場論條目。試圖將所有目標畢其功於一役的理論，請見萬有理論條目。
目录 |
[编辑] 動機
建立新的引力理論的動機隨著年代不同，最早先的動機是要解釋行星軌道（牛頓引力）以及更複雜的軌道（例如：拉格朗日）。再來登場的是不成功的嚐試——要合併引力與波理論或微粒(corpuscular)理論的新引力理論。隨著洛侖茲变换的發現，物理學的樣貌徹底改變，而導致了將其與引力調和的嚐試。在此同時，實驗物理學家開始測試引力與相對論的基礎——洛侖茲不變性、引力造成的光線偏折、Eötvös實驗。這些考量導致與考驗了廣義相對論的發展。
After that, motivations differ. Two major concerns were the development of quantum theory and the discovery of the strong and weak nuclear forces. Attempts to quantize and unify gravity are outside the scope of this article, and so far none has been completely successful.
After general relativity (GR), attempts were made to either improve on theories developed before GR, or to improve GR itself. Many different strategies were attempted, for example the addition of spin to GR, combining a GR-like metric with a space-time that is static with respect to the expansion of the universe, getting extra freedom by adding another parameter. At least one theory was motivated by the desire to develop an alternative to GR that is completely free from singularities.
Experimental tests improved along with the theories. Many of the different strategies that were developed soon after GR were abandoned, and there was a push to develop more general forms of the theories that survived, so that a theory would be ready the moment any test showed a disagreement with GR.
By the 1980s, the increasing accuracy of experimental tests had all led to confirmation of GR, no competitors were left except for those that included GR as a special case, and they can be rejected on the grounds of Occam's Razor until an experimental discrepancy shows up. Further, shortly after that, theorists switched to string theory which was starting to look promising. In the mid 1980s a few experiments were suggesting that gravity was being modified by the addition of a fifth force (or, in one case, of a fifth, sixth and seventh force) acting on the scale of metres. Subsequent experiments eliminated these.
Motivations for the more recent alternative theories 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 GR at the present epoch but may have been quite different in the early universe. Investigation of the Pioneer anomaly has caused renewed public interest in alternatives to General Relativity, but the Pioneer anomaly is too strong to be explained by any such theory of gravity.
[编辑] 本文中的符號標記
為光速，
為引力常數。幾何變數(Geometric variables)在此不使用。
拉丁字母指標取值從1到3，希臘字母指標取值從0到3。採用愛因斯坦取和原則。
為閔可夫斯基度規。
為一張量，通常是度規張量。其有標記(signature)( − , + , + , + )。
共变微分(Covariant differentiation)寫為
或
。
也可考慮閱讀廣義相對論的數學條目。
[编辑] 理論分類
引力理論可以粗略分為數個大類。此處描述的多數理論具有：
若一理論具有一拉格朗日密度，寫作
，則作用量
則是此項的積分，例如： 
其中
是空間的曲率。在此方程中，通常會有
的情形，但並非必要條件。
本文中所描述的理論幾乎每個都有一作用量。這是目前已知的方法來保證能量、動量與角動量守恆能自動成立；儘管如此，要建構使守恆律被違背的作用量仍相當容易。1983年原始版本的MOND並沒有作用量。
一些理論有作用量但沒有拉格朗日密度。一個好的例子是懷海德(1922年)的理論，此中的作用量是非局部的。
一個引力理論是一度規理論(metric theory)僅當其可以給出遵守如下兩個條件的數學表述：
條件1. 存在一度規張量
，標記為1，而此度規掌控了原長(proper-length)與原時(proper-time)測量，一如在狹義與廣義相對論：
此式中對指標μ與ν進行取和。
條件2. 受到引力作用的具應力物質與場按照下列方程反應：
其中
為應力-能量張量，針對所有物質以及非引力的場，而
為隨度規所做的共变導數(covariant derivative)]。
任何引力理論若
永遠成立，則其非度規理論，但任何度規理論可以給予違背條件1與2的數學描述。
度規理論包括（從簡單至複雜）：
- 标量場理論（包括共形平直理論(Conformally flat theories)，以及具有共形平直空間切面(Conformally flat space slices)的層狀理論(Stratified theories)）
诺斯特朗姆(Nordström)、Einstein-Fokker、Whitrow-Morduch、Littlewood、Bergman、Page-Tupper, 愛因斯坦（1912年）、Whitrow-Morduch、羅森(Rosen)（1971年）、Papapetrou、倪維斗(Ni)、Yilmaz、[Coleman]、李-萊特曼-倪(Lee-Lightman-Ni)
羅森（1975年）、Rastall、萊特曼-李(Lightman-Lee)
- 類線性理論（包括線性固定規範(Linear fixed gauge)）
懷海德(Whitehead)、Deser-Laurent、Bollini-Giambini-Tiomno
愛因斯坦廣義相對論
Thiry、Jordan、Brans-Dicke、Bergmann、Wagoner、Nordtvedt、Beckenstein
Will-Nordtvedt、Hellings-Nordvedt
（參見後文1980年代至今的現代理論）
非度規理論，則包括嘉當(Cartan)、Belinfante-Swihart。
關於馬赫原理，在這裡做一些陳述是洽當的，因為其中一些理論根據的是馬赫原理，例如懷海德（1922年），and many mention it in passing eg. Einstein-Grossmann (1913), Brans-Dicke (1961). 馬赫原理可以被想作是介於牛頓與愛因斯坦之間的妥協(half-way-house)。可以做如此描述[1]：
- 牛頓：絕對空間與時間。
- 馬赫：參考系源自於宇宙中物質的分布。
- 愛因斯坦：沒有絕對的參考系。
目前為止，所有的實驗證據指出馬赫原理是不正確的，但其可能性尚未被完全排除。
[编辑] 早期理論（1686年至1916年）
- 另見：廣義相對論的歷史
早期引力理論——指的是廣義相對論之前的理論——包括有牛頓（1686年）、愛因斯坦（1912年a & b）、愛因斯坦與格羅斯曼(Grossmann)（1913年）、诺斯特朗姆(Nordström)（1912年、 1913年）以及愛因斯坦與佛克(Fokker)（1914年）。
在牛頓（1686年）理論中（以更近代的數學重寫），質量密度
產生了一個标量場
：
。
利用倒三角算符(Nabla operator)，可以很方面地寫成：
。
而标量場掌控了自由下落粒子的運動：
。
其中标量場為 
。
The theory of Newton, and Lagrange's improvement on the calculation (applying the variational principle)), completely fails to take into account relativistic effects of course, and so can be rejected as a viable theory of gravity. Even so, Newton's theory is thought to be exactly correct in the limit of weak gravitational fields and low speeds and all other theories of gravity need to reproduce Newton's theory in the appropriate limits.
Also some mechanical explanations of gravitation (incl. Le Sage's theory) were created between 1650 and 1900, but they were overthrown because most of them lead to an unacceptable amount of drag, which is not observed. Other models are violating the energy conservation law and are incompatible with modern thermodynamics.
Einstein's two part publication in 1912 is really only important for historical reasons. By then he knew of the gravitational redshift and the deflection of light. He had realized that Lorentz transformations are not generally applicable, but retained them. The theory states that the speed of light is constant in free space but varies in the presence of matter. The theory was only expected to hold when the source of the gravitational field is stationary. It includes the principle of least action:
where 
is the Minkowski metric, and there is a summation from 1 to 4 over indices 
and 
.
Einstein and Grossmann (1913) includes Riemannian geometry and tensor calculus.
The equations of electrodynamics exactly match those of GR. The equation
is not in GR. It expresses the stress-energy tensor as a function of the matter density.
While this was going on, Abraham was developing an alternative model of gravity in which the speed of light depends on the gravitational field strength and so is variable almost everywhere. Abraham's 1914 review of gravitation models is said to be excellent, but his own model was poor.
The first approach of Nordström (1912) was to retain the Minkowski metric and a constant value of 
but to let mass depend on the gravitational field strength . Allowing this field strength to satisfy
where is rest mass energy and 
is the d'Alembertian,
and
where 
is the four-velocity and the dot is a differential with respect to time.
The second approach of Nordström (1913) is remembered as the first logically consistent relativistic field theory of gravitation ever formulated. From (note, notation of Pais (1982) not Nordström):
where 
is a scalar field,
This theory is Lorentz invariant, satisfies the conservation laws, correctly reduces to the Newtonian limit and satisfies the weak equivalence principle.
Einstein and Fokker (1914) is Einstein's first treatment of gravitation in which general covariance is strictly obeyed. Writing:
they relate Einstein-Grossmann (1913) to Nordström (1913). They also state:
That is, the trace of the stress energy tensor is proportional to the curvature of space.
Einstein (1916, 1917) is what we now know of as General Relativity. Discarding the Minkowski metric entirely, Einstein gets:
}-
which can also be written
}-
Five days before Einstein presented the last equation above, Hilbert had submitted a paper containing an almost identical equation. See relativity priority dispute. Hilbert was the first to correctly state the Einstein-Hilbert action for GR, which is:
where 
is Newton's gravitational constant, 
is the Ricci curvature of space, 
and 
is the action due to mass.
GR is a tensor theory, the equations all contain tensors. Nordström's theories, on the other hand, are scalar theories because the gravitational field is a scalar. Later in this article you will see scalar-tensor theories that contain a scalar field in addition to the tensors of GR, and other variants containing vector fields as well have been developed recently.
[编辑] 自1917年至1980年代的理論
This section includes alternatives to GR published after GR but before the observations of galaxy rotation that led to the hypothesis of "dark matter".
Those considered here include (see Will (1981)[2], Lang (2002)[3]):
Listed by date (the hyperlinks take you further down this article)
Whitehead (1922), Cartan (1922, 1923), Fierz & Pauli (1939), Birkhov (1943), Milne (1948), Thiry (1948), Papapetrou (1954a, 1954b), Littlewood (1953), Jordan (1955), Bergman (1956), Belinfante & Swihart (1957), Yilmaz (1958, 1973), Brans & Dicke (1961), Whitrow & Morduch (1960, 1965), Kustaanheimo (1966) , Kustaanheimo & Nuotio (1967), Deser & Laurent (1968), Page & Tupper (1968), Bergmann (1968), Bollini-Giambini-Tiomno (1970), Nordtveldt (1970), Wagoner (1970), Rosen (1971, 1975, 1975), Ni (1972, 1973), Will & Nordtveldt (1972), Hellings & Nordtveldt (1973), Lightman & Lee (1973), Lee, Lightman & Ni (1974), Beckenstein (1977), Barker (1978), Rastall (1979)
These theories are presented here without a cosmological constant, how to add a cosmological constant or quintessence is discussed under Modern Theories (see also here) or added scalar or vector potential unless specifically noted, for the simple reason that the need for one or both of these was not recognised before the supernova observations by Perlmutter.
[编辑] 标量場論
（參見标量引力理論）
The scalar field theories of Nordström (1912, 1913) have already been discussed. Those of Littlewood (1953), Bergman (1956), Yilmaz (1958), Whitrow and Morduch (1960, 1965) and Page and Tupper (1968) follow the general formula give by Page and Tupper.
According to Page and Tupper (1968), who discuss all these except Nordström (1913), the general scalar field theory comes from the principle of least action:
where the scalar field is,
and may or may not depend on .
In Nordström (1912),
; 
In Littlewood (1953) and Bergmann (1956),
;
In Whitrow and Morduch (1960),
; 
In Whitrow and Morduch (1965),
;
In Page and Tupper (1968),
; 
Page and Tupper (1968) matches Yilmaz (1958) (see also Yilmaz theory of gravitation) to second order when 
.
The gravitational deflection of light has to be zero when c is constant. Given that variable c and zero deflection of light are both in conflict with experiment, the prospect for a successful scalar theory of gravity looks very unlikely. Further, if the parameters of a scalar theory are adjusted so that the deflection of light is correct then the gravitational redshift is likely to be wrong.
Ni (1972) summarised some theories and also created two more. In the first, a pre-existing special relativity space-time and universal time coordinate acts with matter and non-gravitational fields to generate a scalar field. This scalar field acts together with all the rest to generate the metric.
作用量為： 
Misner et al. (1973) gives this without the 
term. is the matter action.
![\Box\phi=4\pi T^{\mu\nu}[\eta_{\mu\nu}e^{-2\phi}+(e^{2\phi}+e^{-2\phi})\partial_\mu t\partial_\nu t]\,](/images/math/8/7/f/87f947a34f9f61087566636716506513.png)
is the universal time coordinate. This theory is self-consistent and complete. But the motion of the solar system through the universe leads to serious disagreement with experiment.
In the second theory of Ni (1972) there are two arbitrary functions 
and 
that are related to the metric by:
![ds^2=e^{-2f(\phi)}dt^2-e^{2f(\phi)}[dx^2+dy^2+dz^2]\,](/images/math/e/1/a/e1a4043872100b7d2cf6f28e4798baad.png)
Ni (1972) quotes Rosen (1971) as having two scalar fields and that are related to the metric by:
![ds^2=\phi^2dt^2-\psi^2[dx^2+dy^2+dz^2]\,](/images/math/4/c/9/4c9be79f3d95df8e6a7e3a4284ddea5e.png)
In Papapetrou (1954a) the gravitational part of the Lagrangian is:
In Papapetrou (1954b) there is a second scalar field 
. The gravitational part of the Lagrangian is now:
[编辑] 雙度規理論
（參見雙度規理論）
雙度規理論 contain both the normal tensor metric and the Minkowski metric (or a metric of constant curvature), and may contain other scalar or vector fields.
Rosen (1973, 1975) Bimetric Theory The action is:
where the vertical line "|" denotes covariant derivative with respect to 
. The field equations may be written in the form:
}-
Lightman-Lee (1973) developed a metric theory based on the non-metric theory of Belinfante and Swihart (1957a, 1957b). The result is known as BSLL theory. Given a tensor field 
, 
, and two constants 
and 
the action is:
and the stress-energy tensor comes from:
In Rastall (1979), the metric is an algebraic function of the Minkowski metric and a Vector field[2]. The Action is:
where
and 
(see Will (1981) for the field equation for 
and 
).
[编辑] 類線性理論
In Whitehead (1922), the physical metric 
is constructed algebraically from the Minkowski metric 
and matter variables, so it doesn't even have a scalar field. The construction is:
![g_{\mu\nu}(x^\alpha)=\eta_{\mu\nu2\int_{\Sigma^-}{y_\mu^- y_\nu^-\over(w^-)^3} [\sqrt{-g}\rho u^\alpha d\Sigma_\alpha]^-](/images/math/a/4/0/a401030fe31a980fe4445fafa033d6c0.png)
}-
where the superscript (-) indicates quantities evaluated along the past light cone of the field point 
and
, 
, 
, 
, 
Deser and Laurent (1968) and Bollini-Giambini-Tiomno (1970) are Linear Fixed Gauge (LFG) 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) 
to define
The action is:
![S={1\over 16\pi G} \int d^4 x\sqrt{-\eta}[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}]+S_m\;](/images/math/d/9/2/d926bcae674f71cb7444b3b05798df92.png)
The Bianchi identity associated with this partial gauge invariance is wrong. LFG theories seek to remedy this by breaking the gauge invariance of the gravitational action through the introduction of auxiliary gravitational fields that couple to .
[编辑] 張量理論
There is only one theory that possesses a single gravitational field that is the metric tensor. Of course this theory is General Relativity.
[编辑] 标量-張量理論
(see also Scalar-tensor theory and Brans-Dicke theory)
These all contain at least one free parameter, as opposed to GR 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 (1948).
Scalar-Tensor theories include Thiry (1948), Jordan (1955), Brans and Dicke (1961), Bergman (1968), Nordtveldt (1970), Wagoner (1970), Bekenstein (1977) and Barker (1978).
The action 
is based on the integral of the Lagrangian 
.
where 
is a different dimensionless function for each different scalar-tensor theory. The function 
plays the same role as the cosmological constant in GR. 
is a dimensionless normalization constant that fixes the present-day value of . An arbitrary potential can be added for the scalar.
The full version is retained in Bergman (1968) and Wagoner (1970). Special cases are:
Nordtvedt (1970), 
I'm not worrying about λ until later in the article, it's discussed under Cosmological Constant.
Brans-Dicke (1961), 
is constant
Bekenstein (1977) Variable Mass Theory Starting with parameters 
and 
, found from a cosmological solution, ![\phi=[1-qf(\phi)]f(\phi)^{-r}\;](/images/math/a/7/7/a77d5579c388e27fae9f4a0dabd9b500.png)
determines function 
then
![\omega(\phi)=-\textstyle\frac{3}{2\textstyle\frac{1}{4}f(\phi)[(1-6q)qf(\phi)-1] [r+(1-r)qf(\phi)]^{-2}\;](/images/math/7/5/d/75dd4b5136ec0457b2be74d1116d5e45.png)
}-
Barker (1978) Constant G Theory
Adjustment of allows Scalar Tensor Theories to tend to GR in the limit of 
in the current epoch. However, there could be significant differences from GR in the early universe.
So long as GR is confirmed by experiment, general Scalar-Tensor theories (including Brans-Dicke) can never be ruled out entirely, but as experiments continue to confirm GR more precisely and the parameters have to be fine-tuned so that the predictions more closely match those of GR.
[编辑] 矢量-張量理論
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 (1973) and Will and Nordtvedt (1972) are both vector-tensor theories. In addition to the metric tensor there is a timelike vector field . The gravitational action is:
![S={1\over 16\pi G}\int d^4 x\sqrt{-g}[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}]+S_m\;](/images/math/0/2/5/025e121d8c08d9d74a470e4877403eec.png)
}-
where , , 
and 
are constants and
}-
See Will (1981) for the field equations for and .
Will and Nordtvedt (1972) is a special case where
; 
Hellings and Nordtvedt (1973) is a special case where
; 
; 
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 they reduce to GR so, so long as GR is confirmed by experiment, general vector-tensor theories can never be ruled out.
[编辑] 其他度規理論
Others metric theories have been proposed; that of Beckenstein (2004) is discussed under Modern Theories.
[编辑] 非度規理論
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 (1981) claims that all non-metric theories are eliminated by Einstein's Equivalence Principle (EEP). Will (2001) tempers that by explaining experimental criteria for testing non-metric theories against EEP. Misner et al. (1973) claims that Cartan's theory is the only non-metric theory to survive all experimental tests up to that date and Turyshev (2006) 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 (1972).
Cartan (1922, 1923) 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 and by Heyl in the years 1958 to 1966.
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 GR, the Lagrangian is made up of a massless and a mass part. The Lagrangian for the massless part is:
The 
is the linear connection. 
is not related in any way to Minkowski but is derived from a completely antisymmetric pseudo-tensor 
as follows:
and 
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:
The space curvature is not Riemannian, but on a Riemannian space-time the Lagrangian would reduce to the Lagrangian of GR.
Some equations of the non-metric theory of Belinfante and Swihart (1957a, 1957b) have already been discussed in the section on bimetric theories.
[编辑] 廣義相對論替代理論的測試
理論與測試的發展是一個牽一個地進行著。多數測試可以被分類為（參見Will 2001）：
- 基本生存力(Basic Viability)
- 愛因斯坦等效原理(Einstein's Equivalence Principle, EEP)
- 參數化後牛頓形式(Parametric Post-Newtonian, PPN)
- 強場引力(Strong Gravity)
- 引力波(Gravitational Waves)
[编辑] 未通過基本生存測試的理論
For more details see Misner et al. (1973) Ch.39 and Will (1981) Table 2.1.
Not all theories of gravity are created equal. Very few, among the multitude in the literature, are sufficiently viable to be worth comparison with General Relativity.
In the early 1970s a group of people at Caltech including Thorne, Will and Ni (see Ni (1972)) compiled a list of twentieth-century theories of gravity. Of each theory they asked the following questions: (i) is it self-consistent? (ii) is it complete? (iii) does it agree, to within several standard deviations, with all experiments performed to date? If a theory failed these criteria they did not always reject it out of hand. If the theory was incomplete in its initial statement, they sometimes tried to complete it by making minor modifications, usually by insisting that in the absence of gravity the laws of physics reverted to those of Special Relativity. For seven of the theories here, the density could be calculated from either 
or 
is the trace of 
; both were considered. Even if that failed, eg. consider the case of Thirry (1948) and Jordan (1955). These are incomplete unless Jordan's parameter is set to -1, in which case they match the theory of Brans-Dicke (1961) and so are worthy of further consideration.
For this section of the present article, the criterion of "all experiments performed to date" is replaced by "with the gross features of Newtonian Mechanics and Special Relativity". Subtler problems are discussed later.
Self-consistency among non-metric theories includes eliminating theories allowing tachyons, ghost poles, 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 (1939); 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, 1973) 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.
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. Milne (1948) is incomplete because it makes no gravitational red-shift prediction.
The theories of Whitrow and Morduch (1960, 1965), Kustaanheimo (1966) and Kustaanheimo and Nuotio (1967) 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 twice that of GR) but light as waves is not.
As an example of disagreement with Newtonian experiments, Birkhoff (1943) theory predicts relativistic effects fairly reliably but demands that sound waves travel at the speed of light, which disagrees violently with experiment.
A modern example of the lack of a relativistic component is MOND by Milgrom, more on that later.
[编辑] 愛因斯坦等效原理(Equivalence Principle, EEP)
等效原理有三個成分：
The first is the uniqueness of free fall, also known as the Weak Equivalence Principle (WEP). This is satisfied if inertial mass is equal to gravitational mass. is a parameter used to test the maximum allowable violation of the WEP. The first tests of the WEP were done by Eötvös before 1900 and limited to less than 5e-9. Modern tests have reduced that to less than 5e-13.
The second is Lorentz invariance. In the absence of gravitational effects the speed of light is constant. The test parameter for this is 
. The first tests of Lorentz invariance were done by Michelson and Morley before 1890 and limited to less than 5e-3. Modern tests have reduced this to less than 1e-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. Spacial local position invariance is tested using gravitational redshift measurements. The test parameter for this is 
. Upper limits on this found by Pound and Rebka in 1960 limited to less than 0.1. Modern tests have reduced this to less than 1e-4.
Schiff's conjecture states that any complete, self-consistent theory of gravity that embodies the WEP necessarily embodies EEP. This is likely to be true if the theory has full energy conservation.
One method for testing EEP is called 
formalism. This is beyond the scope of this article but in a static spherically symmetric gravitational field EEP is satisfied if 
.
Theories of gravity may be metric or non-metric. In metric theories, space-time is endowed with a symmetric metric (usually written ) and trajectories of freely falling bodies are geodesics of that metric. Metric theories satisfy the Einstein Equivalence Principle. Extremely few non-metric theories, if any, satisfy this.
One theory eliminated by formalism is the non-metric theory of Belinfante & Swihart (1957).
[编辑] 參數化後牛頓(Parametric Post-Newtonian, PPN)形式
See also Tests of general relativity, Misner et al. (1973) and Will (1981) for more information.
Work on developing a standard rather than ad-hoc set of tests for evaluating alternative gravitation models began with Eddington in 1922 and resulted in a standard set of PPN numbers in Nordtvedt and Will (1972) and Will and Nordtvedt (1972). Each parameter is a measure of how much a theory departs from Newtonian gravity in a different way. Because we're talking about deviation form Newtonian theory here, this only a measure of weak-field effects. The effects of strong gravitational fields are examined later.
These ten are called : 
, 
, , 
, 
, 
, 
, 
, 
, 
is a measure of space curvature, being zero for Newtonian gravity and one for GR.
is a measure of nonlinearity in the addition of gravitational fields, one for GR.
is a check for preferred location effects.
, , measure the extent and nature of "preferred-frame effects". Any theory of gravity with at least one α nonzero is called a preferred-frame theory.
, , , , 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.
[编辑] 強場引力與引力波
PPN 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 GR.
GR predicts that gravitational waves travel at the speed of light. Many alternatives to GR say that gravitational waves travel faster than light. If true, this could result in failure of causality.
[编辑] 宇宙學測試
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 GR, 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 GR that aim to replace dark matter or dark energy will have to explain why.
[编辑] 理論測試結果
[编辑] 一些理論的PPN參數實測值
（細節參見威爾(Will)（1981年）與倪維斗(Ni)（1972年）。米斯納(Misner)等人（1973年）製表將倪氏參數記號变换成威爾的版本。）
廣義相對論至今已經超過90歲，而不斷繼起的引力替代理論卻無法與更精確的觀測結果相一致。更細節的描述請見參數化後牛頓形式(Parameterized post-Newtonian formalism, PPN)。
下表列舉了為數眾多的理論之PPN值。如果格中的值跟行頂格子的值相同，則表示完整的的式子太複雜而無法列在此處；例如：行頂格子為β參數，而Bergmann（1968年）, Wagoner（1970年）的格子值也是β。
| γ | β | ξ | α1 | α2 | α3 | ζ1 | ζ2 | ζ3 | ζ4 | |
|---|---|---|---|---|---|---|---|---|---|---|
| 愛因斯坦（1916年） 廣義相對論 |
1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 标量-張量理論 | ||||||||||
| Bergmann（1968年）, Wagoner（1970年） |  |
β | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| NordtVedt（1970年）, Bekenstein（1977年） | |
β | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 布蘭斯-狄基（1961年） | |
1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 矢量-張量理論 | γ | β | 0 | α1 | α2 | 0 | 0 | 0 | 0 | 0 |
| Hellings-Nordtvedt（1973年） | γ | β | 0 | α1 | α2 | 0 | 0 | 0 | 0 | 0 |
| Will-Nordtvedt（1972年） | 1 | 1 | 0 | 0 | α2 | 0 | 0 | 0 | 0 | 0 |
| 雙度規理論 | ||||||||||
| Rosen（1975年） | 1 | 1 | 0 | 0 | c0 / c1 − 1 | 0 | 0 | 0 | 0 | 0 |
| Rastall（1979年） | 1 | 1 | 0 | 0 | α2 | 0 | 0 | 0 | 0 | 0 |
| 萊特曼-李（1973年） | γ | β | 0 | α1 | α2 | 0 | 0 | 0 | 0 | 0 |
| 層狀理論 | ||||||||||
| 李-萊特曼-倪（1974年） | ac0 / c1 | β | ξ | α1 | α2 | 0 | 0 | 0 | 0 | 0 |
| 倪維斗（1973年） | ac0 / c1 | bc0 | 0 | α1 | α2 | 0 | 0 | 0 | 0 | 0 |
| 标量場論 | ||||||||||
| 愛因斯坦（1912年）{非廣義相對論} | 0 | 0 | -4 | 0 | -2 | 0 | -1 | 0 | 0† | |
| Whitrow-Morduch（1965年） | 0 | -1 | -4 | 0 | 0 | 0 | -3 | 0 | 0† | |
| 羅森（1971年） | λ |  |
− 4 − 4λ | 0 | -4 | 0 | -1 | 0 | 0 | |
| Papetrou (1954年a, 1954年b) | 1 | 1 | -8 | -4 | 0 | 0 | 2 | 0 | 0 | |
| 倪維斗（1972年）（層狀） | 1 | 1 | -8 | 0 | 0 | 0 | 2 | 0 | 0 | |
| Yilmaz（1958年、1962年） | 1 | 1 | -8 | 0 | -4 | 0 | -2 | 0 | -1† | |
| Page-Tupper（1968年） | γ | β | − 4 − 4γ | 0 | − 2 − 2γ | 0 | ζ2 | 0 | ζ4 | |
| 诺斯特朗姆（1912年） | − 1 |  |
0 | 0 | 0 | 0 | 0 | 0 | 0† | |
| 诺斯特朗姆（1913年）、愛因斯坦-佛克（1914年） | − 1 | |
0 | 0 | 0 | 0 | 0 | 0 | 0 | |
| 倪維斗(Ni)（1972年）（平直） | − 1 | 1 − q | 0 | 0 | 0 | 0 | ζ2 | 0 | 0† | |
| Whitrow-Morduch（1960年） | − 1 | 1 − q | 0 | 0 | 0 | 0 | q | 0 | 0† | |
| Littlewood（1953年）、Bergman(1956年） | − 1 | |
0 | 0 | 0 | 0 | -1 | 0 | 0† |
† 此理論不完備，and ζ4可以是兩值中的一者。最接近零的值在此列出。
至今所有實驗測試與廣義相對論相符，因此PPN分析立即刪除了表中所有的标量場論。
此處未有針對懷海德（1922年）、Deser-Laurent（1968年）、Bollini-Giamiago-Tiomino（1970年）三者的完整PPN參數列表。但在這些三個情形中β = γ，這與廣義相對論的情形以及實驗結果嚴重違背。特別的是，這些理論預測的地球潮汐振幅是不正確的值。
[编辑] 未通過其他測試的理論
非度規理論，例如Belinfante and Swihart (1957a, 1957b)未能與愛因斯坦等效原理的實驗測試相符。
倪維斗的層狀理論（1973年）、李-萊特曼-倪理論（1974年）皆無法解釋水星近日點進動的數據。
The bimetric theories of Lightman and Lee (1973), Rosen (1975), Rastall (1979) all fail some of the tests associated with strong gravitational fields.
The scalar-tensor theories include GR as a special case, but only agree with the PPN values of GR when they are equal to GR. As experimental tests get more accurate, the deviation of the scalar-tensor theories from GR is being squashed to zero.
The same is true of vector-tensor theories, the deviation of the vector-tensor theories from GR is being squashed to zero. Further, vector-tensor theories are semi-conservative; they have a nonzero value for α2 which can have a measurable effect on the Earth's tides.
And that leaves, as a likely valid alternative to GR, nothing [except possibly Cartan (1922), which may violate EEP].
That was the situation until cosmological discoveries pushed the development of modern alternatives.
[编辑] 1980年代至今的現代理論
This section includes alternatives to GR 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: Beckenstein (2004), Moffat (1995), Moffat (2002), Moffat (2005a, b).
These theories are presented with a cosmological constant or added scalar or vector potential.
[编辑] 動機
Motivations for the more recent alternatives to GR 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 GR at the present epoch but may have been quite different in the early universe.
There was a slow dawning realisation in the physics world that there were several problems inherent in the then big bang scenario, two of these were 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. Another alternative was constructing an alternative to GR in which the speed of light was larger 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 of opinion 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 might hold the answer.
The discovery of the accelerated expansion of the universe by Perlmutter 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 GR attempted to explain Rerlmutter's results in a completely different way.
Another observation that sparked recent interest in alternatives to General Relativity is the Pioneer anomaly. It was quickly discovered that alternatives to GR could explain the qualitative features of the anomaly but not the magnitude. Any alternative model of gravity that could explain the Pioneer anomaly would have to depart so strongly from GR that it would fail to satisfy other experimental observations.
[编辑] 宇宙學常數與第五元素(Quintessence)
（參見宇宙學常數、愛因斯坦-希爾伯特作用量、第五元素 (物理學))
宇宙學常數
is a very old idea, going back to Einstein in 1917. The success of the Friedmann model of the universe in which 
led to the general acceptance that it is zero, but the use of a non-zero value came back with a vengeance when Perlmutter discovered 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:
to
In GR, it changes the Einstein-Hilbert action from
to