描述引力波的理论与描述电动力学的电磁波的理论是类似的。

以下内容摘录自维基百科:引力波 – 维基百科,自由的百科全书 (wikipedia.org)

阅读本节需要了解电动力学广义相对论的基本概念,可直接参阅有关书籍[120][121][122][4][123]

线性爱因斯坦方程

引力波——时空的波纹(示意图)

广义相对论预言下的引力波是以波形式传播的时空扰动,被形象地称为“时空涟漪”[124]。广义相对论下的弱引力场可写作对平直时空的线性微扰:(以下采用自然单位引力常数G光速c都设为1)[4]:189-194

,其中

这里是平直时空的闵可夫斯基度规是弱引力场带来的微扰。在这个度规下计算得到的黎曼张量

爱因斯坦张量

这里被称作迹反转度规微扰(trace-reverse metric perturbation)。

如果采用洛伦茨规范,爱因斯坦张量的后三项将为零,这里洛伦茨规范的形式为

事实上总可以选择这样的规范条件,并且洛伦茨规范不是唯一的,意味着坐标在一个无穷小的线性坐标变换下仍满足洛伦茨规范,关于这一点请参考有关规范变换的内容。

在洛伦茨规范下,爱因斯坦张量为

代入爱因斯坦引力场方程

这个方程又叫弱引力场中的线性爱因斯坦方程。在远源()的情形下,得到带有达朗贝尔算符的四维波方程:

引力波的传播

上面波方程的一般解为如下本征函数线性叠加[4]:203-206

其中是四维振幅是四维波矢,满足条件

,这表明引力波传播经过的测地线是零性的,即其传播速度是光速

四维波矢,其中是波的角频率是经典的三维波矢。由于洛伦茨规范并不唯一,此时坐标还不是完全确定的。如果再加上条件:

第一个条件表示引力波张量中所有与时间t有关的分量都为零,第二个条件表示引力波张量矩阵的为零。因此这组规范条件叫做横向无迹规范(transverse traceless gauge),简称TT规范。在TT规范下,。 由洛伦茨规范和TT规范共同决定下的引力波张量只有两个分量是独立的,它们实际对应着引力波的两种偏振态。对于在z方向传播的波矢,这两个振动分量垂直于传播方向,这表明引力波和电磁波一样是横波,其张量形式写作

其中分别为引力波的“十字型”和“交叉型”两种偏振态,上文引力波通过时的效应一节的两幅动画示意了两种偏振各自不同的振动形式。

引力波的辐射

有源的线性爱因斯坦方程解释了波源的运动如何产生引力辐射:

类似用泊松方程求解牛顿引力势,运用格林函数可得到带有推迟势的一般解:[4]:233-234[121]:300-307[39]:第4.1.1节

这里所处在的时间是,表示引力波从源点传播到场点经过了时间为的延迟。

在远场近似和长波极限下,格林函数解近似为

其中标量是源点到场点的距离。

相对论中波源的质能守恒动量守恒合起来写作

因此动量-能量张量中的质量-能量密度)和其他所有和时间t有关的分量(动量密度)对时间的偏导数都为零,代入后方程的解可进一步化简为

这即是引力辐射的四极矩近似公式,描述了一个弱相对论系统引力辐射的最基本情形。其中描述了波源的质量-能量分布

这里张量即是系统的质量四极矩(转动惯量张量),而是波源的质量-能量密度,积分范围是整个波源内部。

四极矩公式的物理意义是引力辐射起始于随时间二阶变化(例如谐振)的四极矩,这一点与电磁辐射不同:电磁辐射起始于随时间二阶变化的偶极矩。这一区别的来源是:一个随时间二阶变化的电偶极矩或磁偶极矩对应着电荷密度中心的振动,这一振动是随意不受限制的;而一个随时间二阶变化的质量的偶极矩对应着质心的振动,这一振动不能满足动量守恒定律,因此不存在这样对时间二阶偏导不为零的质量偶极矩。由于四极矩是偶极矩的更高阶项,这也是引力辐射要远弱于电磁辐射的原因。[125]:第1.2.1节

引力波的能量

四极矩近似下引力波的光度(总辐射功率)为[4]:239-240

 

这里Q是张量矩阵的迹。 引力波的能量通量(单位面积的辐射功率)近似为

这里f是单色引力波的频率。

思考一个地面探测器可以感测到的微弱辐射暴,其频率为1000赫兹,到达地球时的引力强度为10-22的引力波,则其能量通量约为,这相当于满月时地球从月球接收到的电磁辐射能量通量的两倍,大约有1ms之久,这引力波源是夜间天空最亮的星体。这表明引力波实际可以携带很大的能量,但与物质相互作用力非常小,这才是引力波难以被探测的根本原因。[40]:第2.3节

 

 

主要的暗能量的候选者如下:

  • 宇宙学常数,\(\Lambda\)
  • Quintessence,物态方程的参数 \(\omega\) 在 0 到 -1 之间变化,是一个变量
  • Phantom,物态方程的参数 \(\omega\) 可以小于 -1,是一个变量
  • Quintom,物态方程的参数 \(\omega\) 可以等于 -1
  • 修改引力模型?

(238) An Epic Journey to a Black Hole to Give You Goosebumps – YouTube




A black hole is a mysterious place where the laws of physics people are familiar with stop working. Black holes appear when massive stars collapse under their own weight. The gravitational field of the newly formed object is so powerful that even light, including X-rays, can’t escape it. Every black hole has an invisible line-in-the-sand. Cross it – and you won’t be able to escape, even if you’re a beam of light. Beyond the point of no return, the gravity is just too strong. It’s called the event horizon.

There’s a black hole about 1,000 light-years away from the Earth. That’s almost 6 thousand, million, million miles. On the scale of the Universe, it’s right next door. More than four times the mass of our Sun, this medium-sized monster is surrounded by streams of gas. There are two stars nearby. Scientists think this newly discovered black hole might be the nearest to Earth. It’s also the only system with a black hole visible to the unaided eye. Good news – you’re going to visit it right now!

TIMESTAMPS:

  • Nearest black hole to Earth 0:01
  • International Space Station 0:53 🛰
  • The Moon 1:20 🌙
  • Mars 1:36
  • Jupiter 2:02
  • Saturn 2:17
  • The Kuiper Belt 2:43
  • The Oort Cloud 3:14
  • ⚫ You reach your destination! ⚫ 3:50
  • How to see the back of your head 5:01
  • What’s behind the event horizon ❓6:24
  • What happens after spaghettification 🥡 7:16

原文来自:The Black Hole Information Paradox Comes to an End | Quanta Magazine

The Most Famous Paradox in Physics Nears Its End

In a landmark series of calculations, physicists have proved that black holes can shed information, which seems impossible by definition. The work appears to resolve a paradox that Stephen Hawking first described five decades ago.

In a series of breakthrough papers, theoretical physicists have come tantalizingly close to resolving the black hole information paradox that has entranced and bedeviled them for nearly 50 years. Information, they now say with confidence, does escape a black hole. If you jump into one, you will not be gone for good. Particle by particle, the information needed to reconstitute your body will reemerge. Most physicists have long assumed it would; that was the upshot of string theory, their leading candidate for a unified theory of nature. But the new calculations, though inspired by string theory, stand on their own, with nary a string in sight. Information gets out through the workings of gravity itself — just ordinary gravity with a single layer of quantum effects.

This is a peculiar role reversal for gravity. According to Einstein’s general theory of relativity, the gravity of a black hole is so intense that nothing can escape it. The more sophisticated understanding of black holes developed by Stephen Hawking and his colleagues in the 1970s did not question this principle. Hawking and others sought to describe matter in and around black holes using quantum theory, but they continued to describe gravity using Einstein’s classical theory — a hybrid approach that physicists call “semiclassical.” Although the approach predicted new effects at the perimeter of the hole, the interior remained strictly sealed off. Physicists figured that Hawking had nailed the semiclassical calculation. Any further progress would have to treat gravity, too, as quantum.

That is what the authors of the new studies dispute. They have found additional semiclassical effects — new gravitational configurations that Einstein’s theory permits, but that Hawking did not include. Muted at first, these effects come to dominate when the black hole gets to be extremely old. The hole transforms from a hermit kingdom to a vigorously open system. Not only does information spill out, anything new that falls in is regurgitated almost immediately. The revised semiclassical theory has yet to explain how exactly the information gets out, but such has been the pace of discovery in the past two years that theorists already have hints of the escape mechanism.

“That is the most exciting thing that has happened in this subject, I think, since Hawking,” said one of the co-authors, Donald Marolf of the University of California, Santa Barbara.

“It’s a landmark calculation,” said Eva Silverstein of Stanford University, a leading theoretical physicist who was not directly involved.

You might expect the authors to celebrate, but they say they also feel let down. Had the calculation involved deep features of quantum gravity rather than a light dusting, it might have been even harder to pull off, but once that was accomplished, it would have illuminated those depths. So they worry they may have solved this one problem without achieving the broader closure they sought. “The hope was, if we could answer this question — if we could see the information coming out — in order to do that we would have had to learn about the microscopic theory,” said Geoff Penington of the University of California, Berkeley, alluding to a fully quantum theory of gravity.

What it all means is being intensely debated in Zoom calls and webinars. The work is highly mathematical and has a Rube Goldberg quality to it, stringing together one calculational trick after another in a way that is hard to interpret. Wormholes, the holographic principle, emergent space-time, quantum entanglement, quantum computers: Nearly every concept in fundamental physics these days makes an appearance, making the subject both captivating and confounding.

And not everyone is convinced. Some still think that Hawking got it right and that string theory or other novel physics has to come into play if information is to escape. “I’m very resistant to people who come in and say, ‘I’ve got a solution in just quantum mechanics and gravity,’” said Nick Warner of the University of Southern California. “Because it’s taken us around in circles before.”

But almost everyone appears to agree on one thing. In some way or other, space-time itself seems to fall apart at a black hole, implying that space-time is not the root level of reality, but an emergent structure from something deeper. Although Einstein conceived of gravity as the geometry of space-time, his theory also entails the dissolution of space-time, which is ultimately why information can escape its gravitational prison.

The Curve Becomes the Key

In 1992, Don Page and his family spent their Christmas vacation house-sitting in Pasadena, enjoying the swimming pool and watching the Rose Parade. Page, a physicist at the University of Alberta in Canada, also used the break to think about how paradoxical black holes really are. His first studies of black holes, when he was a graduate student in the ’70s, were key to his adviser Stephen Hawking’s realization that black holes emit radiation — the result of random quantum processes at the edge of the hole. Put simply, a black hole rots from the outside in.

 

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本文转载自:

Does Time Really Flow? New Clues Come From a Century-Old Approach to Math.

The laws of physics imply that the passage of time is an illusion. To avoid this conclusion, we might have to rethink the reality of infinitely precise numbers.

If numbers cannot have infinite strings of digits, then the future can never be perfectly preordained.

Dave Whyte for Quanta Magazine

Strangely, although we feel as if we sweep through time on the knife-edge between the fixed past and the open future, that edge — the present — appears nowhere in the existing laws of physics.

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今天在阅读彭罗斯的《通向实在之路》这本书时,看到了33.1节,才知道我一直思考的时空,是“斯奈德-席尔德时空(Snyder-Schild spacetime)”。

本处内容摘录自《通向实在之路》在线电子书,原链接:点击访问

斯奈德-席尔德时空(Snyder-Schild spacetime)出自以下两个论文

这两个参考文献l来自下面这篇文章,专门讨论时空结构:【HEP-TH-9506171】

The Small Scale Structure of Space-Time: A Bibliographical Review (文件大小423KB)

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本文转载自知乎:一文读懂量(xiang)子(ai)纠(xiang)缠(sha)

英文原文1:Entanglement Made Simple

英文原文2:Your Simple (Yes, Simple) Guide to Quantum Entanglement


量子纠缠及其“多世界”诠释都带有一种神秘而迷人的光环。然而,这些都是,或者都应该是科学观点,它们都有实实在在的具体含义。在下面这篇文章中,我们将尽可能简单明了地为大家解释一下量子纠缠和多世界的概念。

纠缠:从经典迈入量子

量子纠缠经常被看作量子力学才独有的现象,但事实并不是这样。实际上,我们可以首先通过思考一个简单的非量子(或者“经典”)现象来考察纠缠,这是一种比较反传统的做法。这样可以让我们绕开量子论中纠缠的怪异之处来体会量子纠缠的精妙。

一个系统由两个子系统组成,纠缠发生在我们对系统的状态有部分了解的情况下。我们将子系统称之为c-on。“c”的意思是“经典的”,为了便于理解,我们把c-on看作蛋糕。

这里我们的蛋糕有两种形状,正方形或者圆形。那么两个蛋糕的总状态就有4种,它们分别是(方,方)(方,圆)(圆,方)(圆,圆)。下面两个表格给出了在四个状态中找到某一个状态的概率。

当我们不能通过一个蛋糕的信息来判断另一个蛋糕的状态时,我们称这两个子系统是独立的。我们的第一个表格就具有这种特性。即使我们知道第一个蛋糕是方的,我们仍然不知道另一个的形状。类似的,第二个子系统的形状并不能告诉我们关于第一个子系统形状的任何有用信息。

另一方面,如果一个蛋糕的信息可以增加我们对另一个蛋糕的认识,我们就说这两个蛋糕是纠缠的。第二个表格中的情况就表现出高度的纠缠。在这种情况中,如果我们已经知道第一个蛋糕是圆的,那么我们就知道第二个蛋糕一定也是圆形的。如果第一个蛋糕是方形的,第二个也是。当我们知道了第一个蛋糕的形状我们就能确定另一个蛋糕的形状。

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Wolfram 语言提供了最先进的全自动向量函数和数据的可视化,用来表示流量、场线和其它任意复杂的矢量场.

流量可视化

StreamPlotListStreamPlot  画出来自矢量场的流量图

LineIntegralConvolutionPlot ▪  ListLineIntegralConvolutionPlot

矢量可视化

VectorPlotListVectorPlot  画出来自矢量场的矢量

向量 + 标量可视化

StreamDensityPlotListStreamDensityPlot  向量场和标量场的叠加图

VectorDensityPlot ▪  ListVectorDensityPlot

3D 向量可视化

VectorPlot3DListVectorPlot3D  根据三维向量场绘制向量

SliceVectorPlot3DListSliceVectorPlot3D  在切片曲面上绘制向量

地理图形可视化 »

GeoVectorPlot ▪  GeoStreamPlot ▪  

复数可视化 »

ComplexVectorPlot ▪  ComplexStreamPlot ▪  

选项

StreamPointsVectorPoints  流线、向量的数或方位

StreamScale  流线的尺度和大小

VectorScalingVectorSizes  向量的尺度和大小

StreamStyleVectorStyle  流线、向量的颜色和样式

StreamMarkersVectorMarkers  流线、向量的一般形状

ColorFunctionMeshMeshFunctions  标量的颜色、网格、等高线等

StreamColorFunction ▪  VectorColorFunction ▪  RegionFunction

VectorAspectRatio ▪  VectorRange ▪  ClippingStyle

转载自:http://www.fourmilab.ch/etexts/einstein/specrel/www/

pdf版本下载:PDF

还有Latex源代码,PostScript格式的下载: LaTeX SourcePostScript


ON THE ELECTRODYNAMICS
OF MOVING BODIES

By A. Einstein
June 30, 1905

It is known that Maxwell’s electrodynamics—as usually understood at the present time—when applied to moving bodies, leads to asymmetries which do not appear to be inherent in the phenomena. Take, for example, the reciprocal electrodynamic action of a magnet and a conductor. The observable phenomenon here depends only on the relative motion of the conductor and the magnet, whereas the customary view draws a sharp distinction between the two cases in which either the one or the other of these bodies is in motion. For if the magnet is in motion and the conductor at rest, there arises in the neighbourhood of the magnet an electric field with a certain definite energy, producing a current at the places where parts of the conductor are situated. But if the magnet is stationary and the conductor in motion, no electric field arises in the neighbourhood of the magnet. In the conductor, however, we find an electromotive force, to which in itself there is no corresponding energy, but which gives rise—assuming equality of relative motion in the two cases discussed—to electric currents of the same path and intensity as those produced by the electric forces in the former case.

Examples of this sort, together with the unsuccessful attempts to discover any motion of the earth relatively to the “light medium,” suggest that the phenomena of electrodynamics as well as of mechanics possess no properties corresponding to the idea of absolute rest. They suggest rather that, as has already been shown to the first order of small quantities, the same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good.1 We will raise this conjecture (the purport of which will hereafter be called the “Principle of Relativity”) to the status of a postulate, and also introduce another postulate, which is only apparently irreconcilable with the former, namely, that light is always propagated in empty space with a definite velocity c which is independent of the state of motion of the emitting body. These two postulates suffice for the attainment of a simple and consistent theory of the electrodynamics of moving bodies based on Maxwell’s theory for stationary bodies. The introduction of a “luminiferous ether” will prove to be superfluous inasmuch as the view here to be developed will not require an “absolutely stationary space” provided with special properties, nor assign a velocity-vector to a point of the empty space in which electromagnetic processes take place.

The theory to be developed is based—like all electrodynamics—on the kinematics of the rigid body, since the assertions of any such theory have to do with the relationships between rigid bodies (systems of co-ordinates), clocks, and electromagnetic processes. Insufficient consideration of this circumstance lies at the root of the difficulties which the electrodynamics of moving bodies at present encounters.

 

I. KINEMATICAL PART

§ 1. Definition of Simultaneity

Let us take a system of co-ordinates in which the equations of Newtonian mechanics hold good.2 In order to render our presentation more precise and to distinguish this system of co-ordinates verbally from others which will be introduced hereafter, we call it the “stationary system.”

If a material point is at rest relatively to this system of co-ordinates, its position can be defined relatively thereto by the employment of rigid standards of measurement and the methods of Euclidean geometry, and can be expressed in Cartesian co-ordinates.

If we wish to describe the motion of a material point, we give the values of its co-ordinates as functions of the time. Now we must bear carefully in mind that a mathematical description of this kind has no physical meaning unless we are quite clear as to what we understand by “time.” We have to take into account that all our judgments in which time plays a part are always judgments of simultaneous events. If, for instance, I say, “That train arrives here at 7 o’clock,” I mean something like this: “The pointing of the small hand of my watch to 7 and the arrival of the train are simultaneous events.”3

It might appear possible to overcome all the difficulties attending the definition of “time” by substituting “the position of the small hand of my watch” for “time.” And in fact such a definition is satisfactory when we are concerned with defining a time exclusively for the place where the watch is located; but it is no longer satisfactory when we have to connect in time series of events occurring at different places, or—what comes to the same thing—to evaluate the times of events occurring at places remote from the watch.

We might, of course, content ourselves with time values determined by an observer stationed together with the watch at the origin of the co-ordinates, and co-ordinating the corresponding positions of the hands with light signals, given out by every event to be timed, and reaching him through empty space. But this co-ordination has the disadvantage that it is not independent of the standpoint of the observer with the watch or clock, as we know from experience. We arrive at a much more practical determination along the following line of thought.

If at the point A of space there is a clock, an observer at A can determine the time values of events in the immediate proximity of A by finding the positions of the hands which are simultaneous with these events. If there is at the point B of space another clock in all respects resembling the one at A, it is possible for an observer at B to determine the time values of events in the immediate neighbourhood of B. But it is not possible without further assumption to compare, in respect of time, an event at A with an event at B. We have so far defined only an “A time” and a “B time.” We have not defined a common “time” for A and B, for the latter cannot be defined at all unless we establish by definition that the “time” required by light to travel from A to B equals the “time” it requires to travel from B to A. Let a ray of light start at the “A time” $t_{\rm A}$from A towards B, let it at the “B time” $t_{\rm B}$ be reflected at B in the direction of A, and arrive again at A at the “A time” $t'_{\rm A}$.

In accordance with definition the two clocks synchronize if

\begin{displaymath}t_{\rm B}-t_{\rm A}=t'_{\rm A}-t_{\rm B}. \end{displaymath}

We assume that this definition of synchronism is free from contradictions, and possible for any number of points; and that the following relations are universally valid:—

  1. If the clock at B synchronizes with the clock at A, the clock at A synchronizes with the clock at B.
  2. If the clock at A synchronizes with the clock at B and also with the clock at C, the clocks at B and C also synchronize with each other.

Thus with the help of certain imaginary physical experiments we have settled what is to be understood by synchronous stationary clocks located at different places, and have evidently obtained a definition of “simultaneous,” or “synchronous,” and of “time.” The “time” of an event is that which is given simultaneously with the event by a stationary clock located at the place of the event, this clock being synchronous, and indeed synchronous for all time determinations, with a specified stationary clock.

In agreement with experience we further assume the quantity

\begin{displaymath}\frac{2{\rm AB}}{t'_A-t_A}=c, \end{displaymath}

to be a universal constant—the velocity of light in empty space.

It is essential to have time defined by means of stationary clocks in the stationary system, and the time now defined being appropriate to the stationary system we call it “the time of the stationary system.”

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