重力波 (相對論)

维基百科，自由的百科全书

此條目指的是相對論中的重力波（-{gravitational wave}-），中国大陸称為引力波，英文中有時也寫作-{gravity wave}-；但更多場合中，-{gravity wave}-是留給地球科學與流體力學中另一種性質迥異的波動，請參見重力波 (流體力學)。另見重力波，消歧義頁。

物理中，在-{zh-cn:引力;zh-tw:重力}-的度規理論，-{zh-cn:引力;zh-tw:重力}-波是時空曲率的擾動以行進波的形式向外傳遞。-{zh-cn:引力;zh-tw:重力}-輻射是另外一種稱呼，指的是這些波從星體或星系中輻射出來的現象。

-{zh-cn:引力波;zh-tw:重力波}-相當微弱，我們所預期在地球上可觀測到的最強-{zh-cn:引力波;zh-tw:重力波}-會來自很遠且古老的事件，在這事件中大量的能量發生劇烈移動（例子包括兩顆中子星的對撞，或兩個極重的黑洞對撞）。這樣的波動會造成地球上各處相對距離的變動，但這些變動的數量級應該頂多-{zh-cn:只;zh-tw:只}-有10²¹分之一。以LIGO-{zh-cn:引力波;zh-tw:重力波}-偵測器的雙臂而言，這樣的變化小於一顆質子直徑的千分之一。這樣的案例應該可以指引出為什麼偵測重力波是十分困難的。

-{zh-cn:引力波;zh-tw:重力波}-的存在而且也真的無所不在，是廣義相對論中一項毫不模糊的預言。所有目前相互競爭而且被“認可”的重力理論（「認可」：與現前可得一切證據能達到相當準確度的相符）所預言的-{zh-cn:引力;zh-tw:重力}-輻射特質即各有千秋；而原則上，這些預言有時候和廣義相對論所預言的相差甚遠。但很不幸地，現在要確認-{zh-cn:引力;zh-tw:重力}-輻射的存在性就已相當具有挑戰性，更不用說要研究它的細節。

雖然-{zh-cn:引力;zh-tw:重力}-輻射並未被清清楚楚地“直接”測到，然而已有顯著的“間接”證據支持它的存在。最著名的是對於脈衝星（或稱波霎）雙星系統PSR1913+16的觀測。這系統被認為具有兩顆中子星，以極其緊密而快速的模式互相環繞對方。其並且呈現了漸進式的旋近(in-spiral)，旋近時率恰好是廣義相對論所預期的值。對於這樣的觀測，最簡單(也幾乎是廣為接受)的解釋為：廣義相對論一定是對這種系統的-{zh-cn:引力;zh-tw:重力}-輻射給出了準確的說明才得以如此。泰勒和赫尔斯因為這些成就共同获得了1993年的诺贝尔物理学奖。

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[编辑] 概觀

在愛因斯坦的廣義相對論裡，重力的本質是時空曲率的表現。約翰·惠勒所推廣的宣傳詞——「物質告訴時空如何彎曲，時空告訴物質如何運動。」——簡單傳神地表達出這項關係。舉例來說，若人站立著，可以感受到地面對足部的壓力。從廣義相對論的觀點，這表示與地面的接觸阻止了物體的自由下落，因而加速了物體。既然加速被視為對世界線的彎折，這表示非在自由下落的人體的世界線並不是短程線(測地線)。另一方面，時空若遠離任何質能則幾乎是完美平直。也因此，短程線表現上近似於較為人知的實體幾何學中的直線，因而小物體可以表現出直線慣性運動。

廣義相對論以及其他類似的重力理論是以寫下場方程式來表達，有時可能另外也寫下運動方程式。也就是說，這些理論是古典相對論性場論，在這之中重力場或多或少都和時空曲率有關。因此某種程度上，一些區域的質能的快速運動會產生時空的漣漪，並向外輻射出，呈現為重力波。換個角度說，這是場的更新資訊從一處向另一處傳遞的表現。

相似於電磁輻射，在廣義相對論（以及其他競爭理論）中，重力輻射以光速前行，並且具有橫波的特性。橫波表示了重力波對於測試粒子運動的影響是發生在與傳播方向相垂直的平面。然而粗略來說：

重力波代表了一個二階張量場的微擾；在量子場論術語中會稱作是「自旋-2」。
電磁波則是來自於向量場的微擾；在量子場論中會稱作是「自旋-1」。
其他類型的物理學波動有很多來自於純量場的微擾；在量子場論中會稱作是「自旋-0」。

在電磁現象中，帶電粒子（例如電子）一些類型的運動會輻射出電磁波。在重力現象中可以作類比，質量或能量一些類型的運動會輻射出重力波。起自於古典電磁學的量子場論稱作量子電動力學，在其之中有種無靜質量的粒子與電磁輻射相應，稱作光子。類比上，我們會想自古典重力理論（廣義相對論）嘗試創造出相應的量子場論，在這樣的理論中會類比地出現一種無靜質量粒子稱作重力子。然而，這種要將廣義相對論量子化的仿效行為，結果發現是失敗的，使得重力子的可能性角色在重力物理學上反而帶來某種程度的麻煩。或許最好的想法是將它視作一種在近似下所得來的概念，而在一些地方這樣的概念有些益處，但無法如同光子的概念可以那樣的健全。

[编辑] 重力波本質

Gravitational waves represent fluctuations in the metric of space-time. That is, they alter the relative distance between test particles. It follows that to directly detect a gravitational wave, you should in essence look for tiny relative motions between two objects. In the case of the LIGO detectors, this is essentially relative motion between two suspended mirrors, and as we saw above the motion to be detected is far smaller than the size of an atom, in fact smaller than the "size" of an atomic nucleus. Since thermal motion in each mirror is far larger than this, understanding why anyone would expect LIGO to work takes some explaining! (See LIGO.)

Imagine a perfect flat region of spacetime, with a bunch of mutually motionless test particles. Along comes a monochromatic linearly polarized gravitational wave. What happens to the test particles? Roughly speaking, they will oscillate in a cruciform manner, orthogonal to the direction of motion:

first, East/West separated particles draw together while North/South separated particles draw apart,
next, East/West separated particles draw apart while North/South separated particles draw together,

and so forth. (Diagonally separated particles exhibit a relative motion which is more difficult to describe verbally, but which is more or less implied by this description.) The cross-sectional of a small box of test particles is invariant under these changes, and there is neglible motion in the direction of propagation (at least, neglecting gravitomagnetic effects; that is, we are tacitly assuming that the relative motion of our test particles is not very rapid).

A monochromatic circularly polarized induces similar cruciform oscillation, except that the crucifix rotates with the same frequency as the frequency of cruciform oscillation.

Interestingly enough, after the wave has passed, there may be some residual "secular" relative motion of the test particles. There are also some interesting optical effects. If, before the wave arrives, we look through the oncoming wavefronts at objects behind these wavefronts, we can see no optical distortion (if we could, of course, we would have advance notice of its impending arrival, in violation of the principle of causality). But if, after the wave has passed by, we turn and look through the departing wavefronts at objects which the wave has not yet reached, we will see optical distortions in the images of small shapes such as galaxies. Unfortunately, this is an utterly impractical method of detecting the very weak waves we can expect to occur in the vicinity of the solar system.

[编辑] 重力波源

Gravitational waves are caused by certain motions of mass or energy. The type of motion required is different from electromagnetism in one very important respect however: the strongest type of electromagnetic radiation is dipole radiation, while the strongest type of gravitational radiation is quadrupole radiation. [1]

According to general relativity, the quadrupole moment (or some higher moment) of an isolated system must be time-varying in order for it to emit gravitational radiation. Here are some examples which illustrate when we should (assuming general relativity gives accurate predictions) expect a system to emit gravitational radiation:

An isolated object in approximately "rectilinear" motion will not radiate. (Needless to say, this motion is wrt some observer and can be only approximately rectilinear. Technically, this entails defining a weakly gravitating system possessing a time varying dipole moment but stationary quadrupole moment, with all moments being taken with respect to the origin.) This can be regarded as a consequence of the principle of conservation of linear momentum. (Caveat: this example is trickier than it looks, and in the case of a small object falling toward a large one, say, it leads to one of the most vexed questions in general relativity, the problem of treating radiation reaction).

A spherically pulsating spherical star (nonzero and non-stationary monopole moment or mass, but vanishing and hence stationary quadrupole moment) will not radiate, in agreement with Birkhoff's theorem.

A spinning disk (nonzero but stationary monopole and quadrupole moments) will not radiate. This can be regarded as a consequence of the principle of conservation of angular momentum. (Caveat: in general relativity, unlike Newtonian gravitation, a spinning disk will not generate an external field identical to the field of an equivalent but non-spinning disk, due to gravitomagnetic effects, but this does not contradict the absence of radiation. Roughly speaking, the field is generated as we concentrate matter, and if that matter has some angular momentum, but we end up with a stationary external gravitational field, that field will exhibit gravitomagnetism but not radiation.)

Two objects mounted on the endpoints of an isolated extensible curtain rod, which is provided with some kind of engine and which oscillates long/short/long with frequency $ω$ , gives a system with time-varying quadrupole moment, so this system will radiate. Observers far from the rod and in the equatorial plane of the rod will observe linearly polarized radiation (aligned with the rod) with frequency $ω$ . Observers lying on the axis of symmetry of the rod will observe no radiation, however.

A spinning non-axisymmetric planetoid (say with a large bump or dimple on the equator) will define a system with a time-varying quadrupole moment, so this system will radiate. As an idealization, one can study an isolated uniform mass curtain rod which is spinning with angular frequency $ω$ about a rotation axis orthogonal to the rod, but passing through some point other than the centroid of the rod. This gives a system with time varying quadrupole moment, so the system will radiate. Observers far from the system and lying in the plane of rotation will observe linearly polarized radiation with frequency $2ω$ . Observers far from the system and near its axis of symmetry will observe circularly polarized radiation with frequency $ω$ .

Two objects orbiting each other with angular frequency $ω$ in a quasi-Keplerian planar orbit, gives a system with time-varying quadrupole moment, so this system will radiate. Observers far from the system and in its equatorial plane will observe linearly polarized radiation (aligned with the rod) with frequency $2ω$ . Observers far from the system and lying on its axis of symmetry will observe circularly polarized radiation with frequency $ω$ .

The last three examples illustrate a general rule-of-thumb: far from a radiating system, projection of the system on the "viewing plane" affords a rough and ready indication of what kind of radiation will be observed.

These examples (and others) are most commonly studied using a simplified version of general relativity, sometimes called 線性化廣義相對論, which gives indistinguishable results in the case of weak gravitational fields. (The external field of our Sun would be considered "weak" in this terminology.) Similar conclusions hold for the fully nonlinear theory, but it is much more difficult to obtain analytic results outside the domain of the linearized theory. This is one reason why so much work on phenomena such as the collision and merger of two black holes currently requires 數值分析.

Gravitational radiation carries energy away from a radiating system. Consequently, in the case of the quasi-Keplerian system discussed above, the two objects will gradually spiral in towards one another, becoming more tightly bound to compensate for this loss of energy. The predicted rate of this inspiral can also be computed, using the linearized approximation, and the result gives excellent agreement for observed binary pulsars (this is the theoretical basis for the Nobel Prize awarded to Hulse and Taylor). In the late stages of the inspiral of two neutron stars or black holes, however, the linearized theory is no longer adequate, so one must result to more complicated approximations, and eventually to numerical simulations.

Similarly, in the case of the eccentric rotating rod, the frequency will decrease as the radiation gradually carries off energy from the system.

We stress that some theories of gravitation give significantly different predictions concerning the nature and generation of gravitational radiation, while others give predictions which are almost identical to those of general relativity. All currently known theories other than general relativity are either in disagreement with observation, or in some sense more complicated than general relativity (see for example Brans-Dicke theory for an example illustrating the latter possibility).

If two spinning black holes were to collide, they could emit an enormous amount of gravitational radiation and lose energy in the process.

[编辑] 偵測

Russell Alan Hulse and Joseph Hooton Taylor Jr. were awarded the Nobel Prize in Physics in 1993年 for their observations of a remarkable binary pulsar, PSR B1913+16. According to general relativity, this system should emit gravitational radiation which carries off energy at a specific rate, which should in turn cause the orbit to decay at a rate of roughly 7 mm per day. This prediction agrees with the observations of Hulse and Taylor.

But to directly detect gravitational waves you would have to look for any motion they cause. Typically you would look for the expansion and contraction oscillations caused by the gravitational wave. A simple version of this setup is called a 韋伯棒 -- a large, solid piece of metal with electronics attached to detect any vibrations. Unfortunately, Weber bars are not likely to be sensitive enough to detect anything but very powerful gravitational waves. A more sensitive version is the 干涉儀, with test masses placed as many as four kilometers apart. Ground-based interferometers such as LIGO are now coming on line. The motion to be detected would be very slight -- a small fraction of the width of an atom, over a distance of four kilometers. A number of teams are working on making more sensitive and selective gravitational wave detectors and analysing their results. Space-based interferometers, such as LISA are also being developed.

One reason for the lack of direct detection so far is that the gravitational waves that we expect to be produced in nature are very weak, so that the signals for gravitational waves, if they exist, are buried under noise generated from other sources. Reportedly, ordinary terrestrial sources would be undetectable, despite their closeness, because of the great relative weakness of the 重力.

A commonly used technique to reduce the effects of noise is to use coincidence detection to filter out events that do not register on both detectors. There are two common types of detectors used in these experiments:

雷射干涉儀, which use long light paths, such as GEO, LIGO, TAMA, VIRGO, ACIGA and the space-based LISA;
resonant mass gravitational wave detectors which use large masses at very low temperatures such as AURIGA, ALLEGRO, EXPLORER and NAUTILUS.

There are other prospects such as MiniGRAIL, a spherical gravitational wave antenna based at Leiden University. Some scientists even want to use the moon as a giant gravitational wave detector. The moon should be somewhat pliable to the contortions caused by gravitational waves.

[编辑] 「愛因斯坦在你家」重力波搜尋計畫(Einstein@Home)

Bruce Allen of UWM's LIGO Scientific Collaboration (LSC) group is leading the development of the 愛因斯坦在你家計畫(Einstein@Home), developed to search data for signals coming from selected, extremely dense, rapidly rotating stars observed from LIGO in the US and the GEO 600 gravitational wave observatory in Germany . Such sources are believed to be either 夸克星 or 中子星; a subclass of these stars are already observed by conventional means and are known as 波霎, electromagnetic wave-emitting celestial bodies. If some of these stars are not quite near-perfectly spherical, they should emit gravitational waves, which LIGO and GEO 600 may begin to detect.

Einstein@Home is a small part of the LSC scientific program. It has been set up and released as a distributed computing project similar to SETI@home. That is, it relies on computer time donated by private computer users to process data generated by LIGO's and GEO 600's search for gravity waves.

[编辑] 展望

物理學中未解決的問題: 我們宇宙是否充滿始於大爆炸的重力輻射？又或是源於天文物理的客體，例如旋近的中子星？這些現象對於量子重力與廣義相對論方面，能夠告訴我們些什麼呢？

科學家渴望能夠自一些難以或無法利用電磁輻射來偵測的天文客體，直接觀測到重力波，用之來探察現象。舉例來說，雖然黑洞不像一般星體會放出可見的電磁輻射 (另見重力紅移)，然而當一個物體掉入黑洞時，重力波會被發射出來；另外的發射場合是兩個黑洞互撞。If the inspiraling mass is significantly smaller than the central black hole, the emitted gravitational waves may, at least in some circumstances, allow physicists to directly probe the spacetime geometry around the event horizon (such observations are a primary goal of the LISA mission). Also, because gravitational waves are so weak (and thus difficult to detect), objects opaque to light are often transparent to gravitational radiation. In particular, gravitational waves could propagate while the universe was still opaque to light (i.e., at times before recombination). In this way, gravitational waves could help reveal information about the very structure of the 宇宙.

In contrast to electromagnetic radiation, it is not fully understood what difference the presence of gravitational radiation would make for the workings of the universe. A sufficiently strong sea of primordial gravitational radiation, with an energy density exceeding that of the 大霹靂 electromagnetic radiation by a few orders of magnitude, would shorten the life of the universe, violating existing data that show it is at least 13 billion years old. More promising is the hope to detect waves emitted by sources on astronomic size scales, such as:

超新星或伽瑪射線爆；
"chirps" from inspiraling coalescing binary stars;
periodic signals from spherically asymmetric neutron stars or quark stars;
stochastic gravitational wave background sources.

[编辑] 推導

關於以下內容，閱者可能需要具備廣義相對論會用到張量代數、微分幾何等數學基礎。

[编辑] 平直時空的微擾

考慮近乎平直的完整度規 $g\,$ ，而寫成平直度規 $\eta\,$ 加上一些微擾 $h\,$ 。

$g_{\mu \nu} = \eta_{\mu \nu} + h_{\mu \nu} \,$

真空中的愛因斯坦方程式為

$R_{\mu \nu} = \mathbf{0}$

其中 $R\,$ 是芮奇曲率。我們將對 $R\,$ 就 $h\,$ 的冪級做微擾式的展開。

$R_{\mu \nu} = \mathbf{0} + \delta R_{\mu \nu} + \delta^2 R_{\mu \nu} + \cdots$

零階項僅能是平直度規的函數，因此實際值為零。既然微擾該當很小，我們只解開一階項並忽略高階項。

$R_{\mu \nu} = \delta R_{\mu \nu} + \mathbf{0}$

其中 $\delta R_{\mu \nu}\,$ 是對於平直芮奇曲率(因此是零)的偏離，其和微擾 $h\,$ 呈線性關係。

現在我們要用到芮奇曲率的式子：

$R_{\mu \nu} = \partial_{\alpha} \Gamma_{\mu \nu}^\alpha - \partial_{\nu} \Gamma_{\mu \alpha}^\alpha + \Gamma_{\mu \nu}^\alpha \Gamma_{\alpha \beta}^\beta - \Gamma_{\mu \beta}^\alpha \Gamma_{\nu \alpha}^\beta$

其中 $\Gamma\,$ 是克里斯托菲爾符號，而 $\partial_{\alpha}$ 是 $\frac{\partial}{\partial x^{\alpha}}$ 的簡略形。只有前兩項會對一階修正有貢獻，此二在 $\Gamma\,$ 中是線性的：

$\delta R_{\mu \nu} = \partial_{\alpha} \delta \Gamma_{\mu \nu}^\alpha - \partial_{\nu} \delta \Gamma_{\mu \alpha}^\alpha$

接著我們要用到克里斯托菲爾符號的式子：

$\Gamma^\alpha_{\mu \nu} = \frac{1}{2} g^{\alpha \gamma} \left( \partial_{\nu} g_{\gamma \mu} + \partial_{\mu} g_{\gamma \nu} - \partial_{\gamma} g_{\mu \nu} \right)$

將平直度規看作是常數，則僅存的一階項們所牽涉到的會是微擾的導數。

$\delta \Gamma^\alpha_{\mu \nu} = \frac{1}{2} \eta^{\alpha \gamma} \left( \partial_{\nu} h_{\gamma \mu} + \partial_{\mu} h_{\gamma \nu} - \partial_{\gamma} h_{\mu \nu} \right)$

現在，線性化的愛因斯坦方程式變成為：

$\delta R_{\mu \nu} = \frac{1}{2} \left( \Box^2 h_{\mu \nu} + \partial_\alpha V_\beta + \partial_\beta V_\alpha \right)$

其中以 $V_\alpha\,$ 代表 $\partial_\beta h_\alpha^\beta - \frac{1}{2} \partial_\alpha h_\beta^\beta$ ，而 $\Box^2 = \partial_t^2 - \nabla^2$ 是達朗貝爾算符(d'Alembertian)或4-拉普拉斯算符。升降指標可以是很具技巧性的。到第一階，你只需要用到平直度規。此外注意到反度規(inverse metric)帶有負的微擾加上其高階項。

接著，我們挑選了一個特別的座標系統，其中 $V_\alpha\,$ 恰是零。要能如此需做一些證明，不過事實上真的可以。我們得到一條波動方程式與我們的規範條件(gauge condition)。

$\Box^2 h_{\mu \nu} = \mathbf{0}$

$\partial_\beta h_\alpha^\beta = \frac{1}{2} \partial_\alpha h_\beta^\beta$

從更簡單的波動方程式所得的經驗，我們可以猜出解的一般形式：

$h_{\mu \nu} = A_{\mu \nu} e^{\imath k \cdot x}$

其中 $k \cdot k = 0$ ，是個零向量(null vector)。波動方程式現在被滿足了，不過 $A\,$ 該選什麼才能滿足我們所用的規範條件呢？

$A_\alpha^\beta \partial_\beta e^{\imath k \cdot x} = A_\beta^\beta \partial_\alpha e^{\imath k \cdot x}$

$A_\alpha^\beta k_\beta = A_\beta^\beta k_\alpha$

如果我們不希望轉換式會干擾了我們的規範選擇，那麼我們最好讓波是零對角元素和(traceless)： $A_\beta^\beta = 0$ ，且是橫波： $A_\alpha^\beta k_\beta = 0$ 。

對於一個沿 $z\,$ 方向前行的波， $k = (1,0,0,1)\,$ ，微擾會是如下形式：

$h_{\mu \nu} = \begin{bmatrix} 0 & 0 & 0 & 0\\ 0 & A_{+} & A_{\times} & 0\\ 0 & A_{\times} & -A_{+} & 0\\ 0 & 0 & 0 & 0 \end{bmatrix} e^{\imath k \cdot x}$

因此振動會是橫向的空間扭曲。這樣的波稱為自旋-2因為存在有2個不同的偏振。光只有1！ $A_{+}\,$ 稱為加號偏振(plus polarization)而 $A_{\times}\,$ 稱為叉號偏振(cross polarization)。

[编辑] 有源微擾

愛因斯坦方程式描述了時空曲率與造成曲率的物質或源之間的關係：

$R_{\alpha \beta} - \frac{1}{2} g_{\alpha \beta} R = \frac{8 \pi G}{c^4} T_{\alpha \beta}$

我們對曲率的第一階貢獻在前面就已經決定了，

$\delta R_{\mu \nu} = \frac{1}{2} \Box^2 h_{\mu \nu}$

我們現在只關注洛侖茲規範(Lorentz gauge)這個選項，現在可以將它寫成很有意義的形式：

$\frac{\partial}{\partial^\beta} \left( h_{\alpha \beta} - \frac{1}{2} h \eta_{\alpha \beta} \right)$

右手邊是有對角元素反轉(trace-reversed)之 $h_{\alpha \beta}\,$ 的散度。自此以後，對角元素反轉的微擾將簡化為 $\overline{h}_{\alpha \beta}$ 。

我們現在可以將這些方程式併入線性化愛因斯坦方程式：

$\Box^2 \overline{h}_{\alpha \beta} = - \frac{16 \pi G}{c^4} T_{\alpha \beta}$

這可以類比於有源電磁波，在電磁學上是個解決已久的問題，解決的辦法是遲滯格林函數(retarded Green's functions)：

$\overline{h}^{\alpha \beta} \left(t,\vec{x} \right) = \frac{4 G}{c^4} \int d^3y \frac{T^{\alpha \beta}\left(t_r,\vec{y}\right)}{\left| \vec{x}-\vec{y} \right|}$

其中 $t_r = t - \frac{\left| \vec{x}-\vec{y} \right|}{c}$ 是遲滯時間(retarded time)。

[编辑] 遠源的近似

If we want to study metric perturbations far from the source then we can envoke a very useful approximation.

$\overline{h}^{\alpha \beta} \left(t,\vec{x} \right) \approx \frac{4 G}{c^4} \frac{1}{r} \int d^3y T^{\alpha \beta}\left(t - \frac{r}{c},\vec{y}\right)$

Where $r$ is the approximate distance to the source.

We now invoke the local conservation of energy-momentum (to first order) to find useful interrelationships in the stress-energy tensor.

$\nabla \cdot \mathbf{T} = 0$

$\frac{\partial}{\partial t} T^{tt} = -\frac{\partial}{\partial x^i} T^{ti}$

$\frac{\partial^2}{\partial t^2} T^{tt} = - \frac{\partial^2}{\partial t \partial x^i} T^{ti}$

$\frac{\partial}{\partial t} T^{tj} = - \frac{\partial}{\partial x^i} T^{ij}$

$\frac{\partial^2}{\partial t^2} T^{tt} = \frac{\partial^2}{\partial x^i \partial x^j} T^{ij}$

We now take this relationship and massage it into the form of our original integral and see what new information it gives us.

$\int d^3x x^k x^l \frac{\partial^2}{\partial t^2} T^{tt} = \int d^3x x^k x^l \frac{\partial^2}{\partial x^i \partial x^j} T^{ij}$

We wanted to multiply the right hand side with with the two powers of $x$ so that we can integrate by parts twice and get down to a regular volume integral.

$\frac{d^2}{d t^2} \int d^3x x^k x^l T^{tt} = 2 \int d^3x T^{kl}$

Assuming the stress-energy tensor takes the simple form

T αβ = ρ u α u β

Where $ρ$ is the mass density and $u α$ is the 4-velocity. If the source is nonrelativistic, then the energy density will be dominated by the mass density, $T t t = ρ$

$\frac{d^2}{d t^2} \int d^3x x^k x^l \rho = 2 \int d^3x T^{kl}$

Here we see something very similar to the moment of inertia, we call it the second mass moment.

$I^{kl}(t) = \int d^3x x^k x^l \rho\left(t,\vec{x} \right)$

We now have our final expression that relates the gravitational waves with their source.

$\overline{h}^{kl} \left(t,\vec{x} \right) \approx \frac{2 G}{c^4} \frac{1}{r} \frac{d^2}{d t^2} I^{kl} \left(t,\vec{x} \right)$

[编辑] 微擾法對精確解

重力波和電磁波性質迥異，理由是電磁波可以精確地從麥克斯韋方程式推導出來。然而重力波，作為線性、自旋-2的波，常被看作僅僅是特定時空幾何的微擾罷了。換言之，現實中總是會有線性、自旋-1的電磁波，卻不存在有線性、自旋-2的重力波。雖說仍有波樣的擾動，但一般來講，一如廣義相對論中總會出現的，事物是非線性的。這也是重力子可能不存在的理由之一。

[编辑] 重力波會傳遞能量

科學社群中有部分人一開始對於「重力波是否會如同電磁波一般可以傳遞能量」感到困惑，這樣的困惑來自於一項事實：重力波沒有局域能量密度——如此對於應力-能量張量的量值不會造成貢獻。不像牛頓重力，愛因斯坦重力不是一項力理論。重力在廣義相對論中不是一種力，它是幾何。因此這樣的場原來被認為不含能量，一如重力勢。然而這場確實可以攜帶能量，如同它可以在遠處作出機械功。而這已經用可傳輸能量的應力-能量偽張量進行證明過，也可看出輻射是如何將能量往外攜帶到無限遠處。

[编辑] 相關條目

重力磁性(Gravitomagnetism)
重力子
LIGO，美國重力波偵測器。
VIRGO與GEO 600，歐洲的兩處偵測器。
TAMA，日本偵測器。.
LISA，計畫中的雷射干涉儀太空天線，預計2015年升空。
Sticky bead argument，能看出重力輻射應該攜帶能量的費因曼方法。
pp-wave spacetime，利用平面波前來為重力輻射(可能伴隨電磁或其他輻射)建立重要的精確解模型類別。

[编辑] 外部連結

USENET上的物理學問答集(physics FAQ)中的重力輻射討論
重力波偵測器列表
雷射干涉儀重力波觀測站(縮寫LIGO)——加州理工學院LIGO實驗室
Info page for "Einstein@Home," a distributed computing project processing raw data from LIGO Laboratory, at CalTech searching for gravity waves
愛因斯坦在你家計劃首頁
義大利研究者關於EXPLORER與NAUTILUS資料分析之論文
重力波物理中心——美國國家科學基金會 [PHY 01- 14375]
澳大利亞國際重力研究中心——西澳大利亞大學
TAMA計畫——開發針對公里級干涉儀的高等技術。
"Could superconductors transmute electromagnetic radiation into gravitational waves?"——科學美國人(Scientific American)文章
Science to ride gravitational waves, 英國國家廣播公司(BBC)新聞(2005年11月announcement of science run of LIGO and GEO 600 gravitational wave detectors).
[2]

[编辑] 文獻

B. Allen, et al., Observational Limit on Gravitational Waves from Binary Neutron Stars in the Galaxy. The American Physical Society, March 31, 1999.
Gravitational Radiation. Davis Associates, Inc.
Amos, Jonathan, Gravity wave detector all set. BBC, February 28, 2003.
Rickyjames, Doing the (Gravity) Wave. SciScoop, December 8, 2003.
Will, Clifford M., The Confrontation between General Relativity and Experiment. McDonnell Center for the Space Sciences, Department of Physics, Washington University, St. Louis MO.
Chakrabarty, Indrajit, "Gravitational Waves: An Introduction". arXiv:physics/9908041 v1, Aug 21, 1999.

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页面分类: 翻譯請求 | 物理學中未解決的問題 | 相对论 | 天文学

重力波 (相對論)

维基百科，自由的百科全书

目录

[编辑] 概觀

[编辑] 重力波本質

[编辑] 重力波源

[编辑] 偵測

[编辑] 「愛因斯坦在你家」重力波搜尋計畫(Einstein@Home)

[编辑] 展望

[编辑] 推導

[编辑] 平直時空的微擾

[编辑] 有源微擾

[编辑] 遠源的近似

[编辑] 微擾法對精確解

[编辑] 重力波會傳遞能量

[编辑] 相關條目

[编辑] 外部連結

[编辑] 文獻

Views

导航

帮助

搜索

其他语言

重力波 (相對論)

维基百科，自由的百科全书

目录

[编辑] 概觀

[编辑] 重力波本質

[编辑] 重力波源

[编辑] 偵測

[编辑] 「愛因斯坦在你家」重力波搜尋計畫(Einstein@Home)

[编辑] 展望

[编辑] 推導

[编辑] 平直時空的微擾

[编辑] 有源微擾

[编辑] 遠源的近似

[编辑] 微擾法 對 精確解

[编辑] 重力波會傳遞能量

[编辑] 相關條目

[编辑] 外部連結

[编辑] 文獻

Views

导航

帮助

搜索

其他语言

[编辑] 微擾法對精確解