Astronomy from 4 Perspectives 16 Jena 2015

Gravitational waves and cosmic expansion: similarities anddi↵erences

Markus Possel, Haus der Astronomie Heidelberg

Gravitational waves and cosmic expansion are both described in terms of Einstein’s general re-lativity. This article explores the similarities between the two phenomena, as well as somedi↵erences, using the fundamental concept of the metric of spacetime. In both contexts, the fo-cus is on the metric of space, as opposed to the metric components determining the propertiesof time, which allows for a simplified, more accessible discussion.

The main focus of this summer school is ongravitational waves. But in order to under-stand how these waves interact with matter(including gravitational wave detectors), it ishelpful to take a broader view that includesnot only gravitational waves but another phe-nomenon intimately connected with the re-lativistic description of our universe: cosmicexpansion as the basis for our current modelsof cosmology. Both phenomena are describedin terms of the spacetime metric encoding thegeometry of space and time. We begin byintroducing the concept of spacetime metricsover the next three sections.

The three–fold role of coordinates

Usual Cartesian coordinates in physics servea threefold purpose. For one, they providea scheme for naming points in space. Onceyou have introduced your Cartesian coordin-ate system, it is as if every point in space weresporting a little name tag, bearing a uniquename such as (1.0, �2.6, 9.1) or similar. Theunique name can be used to refer to that spe-cific point, and that point only.

In addition, the coordinate values encodesome information about proximity. In theCartesian plane, the point (0, 0.5) is closer tothe point (0, 0.4) than to the point (0, 9.5).Finally, Cartesian coordinates allow for a dir-ect computation of distances between points.Given two points and their coordinates, sayP1 = (x1, y1, z1) and P2 = (x2, y2, z2), theirdistance s is given by

s =p

(x1 � x2)2 + (y1 � y2)2 + (z1 � z2)2,(1)

corresponding to a three-dimensional versionof Pythagoras’ theorem that can be abbrevi-ated as

s2 = �x2 + �y2 + �z2. (2)

An additional bonus is that when we plotpoints in Cartesian coordinates, say the pointsin a plane, we usually make sure to draw alldistances faithfully. Hence, in the usual xycoordinate system, drawn on a piece of paper,we can simply use a ruler to directly measurethe distances between points, and rest assuredthat calculations using point coordinates willyield the same result (bar scale factors linkingthe scale of our drawing and the scale of theruler). This is, of course, a teaching tool thatis used extensively in a school setting.

For more general spaces and surfaces, andeven for non-Cartesian coordinates in regular,Euclidean space or on a Euclidean space (suchas spherical coordinates or polar coordinates),the relationship between distances and co-ordinate values is more complex. A simple,one-dimensional example is that of a row ofhouses, numbered with integers, as sketchedin figure 1.

An additional bonus is that when we plot points in Cartesian coordinates, say the pointsin a plane, we usually make sure to draw all distances faithfully. Hence, in the usualxy coordinate system, drawn on a piece of paper, we can simply use a ruler to directlymeasure the distances between points, and rest assured that calculations using pointcoordinates will yield the same result (bar scale factors linking the scale of our drawingand the scale of the ruler). This is, of course, a teaching tool that is used extensively ina school setting.

For more general spaces and surfaces, and even for non-Cartesian coordinates inregular, Euclidean space or on a Euclidean space (such as spherical coordinates orpolar coordinates), the relationship between distances and coordinate values is morecomplex. A simple, one-dimensional example is that of a row of houses, numberedwith integers, as sketched in figure 1. To begin with, the house numbers are merely

1 2 3 4 5 6 7 8

Figure 1: A row of houses, numbered with integers

markers for each house; in our simple sketch, they have been associated with the cen-ter of the house, determined along the one-dimensional street. We can extend thesenumbers to a one-dimensional coordinate system by subdividing the distance betweenone marker and the next evenly; the point half-way between houses no. 4 and 5, forinstance, would be the point with coordinate 4.5, and the point three-quarters of theway between houses 2 and 3 would have 2.75 as its coordinate.

Coordinates defined in this way perform two of the three roles we identified forCartesian coordinates: they uniquely identify each point along the street, and theyencode proximity: regarding house no. 3, house no. 5 is further away than house no.4.

What the coordinates do not allow is for a direct calculation of distances – at leastnot without additional information. On the other hand, this changes if we have infor-mation about the distances between the houses – how far away is the mark in front ofno. 1 from that in front of no. 2? How far away is the no. 2 mark from the no. 3 mark,and so on? If we know that, say, the no. 4 and no. 5 marks are 6.8 m apart, then weknow that our point 4.5 is 3.4 m from the mark no. 4, and the same distance from theno. 5 mark.

With this additional information, we can use our coordinates to compute distancesbetween points along the street – as long as we have that additional input, the informa-tion about how far the houses are apart. This additional information is our first exampleof a metric: a set of information that allows you to translate coordinate di↵erences into

2

Fig. 1: A row of houses, numbered with in-tegers

To begin with, the house numbers aremerely markers for each house; in our simplesketch, they have been associated with thecenter of the house, determined along the one-dimensional street. We can extend these num-bers to a one-dimensional coordinate systemby subdividing evenly the distance betweenone marker and the next; the point half-way

Astronomy from 4 Perspectives 17 Jena 2015

between houses no. 4 and 5, for instance,would be the point with coordinate 4.5, andthe point three-quarters of the way betweenhouses 2 and 3 would have 2.75 as its coordin-ate.

Coordinates defined in this way performtwo of the three roles we identified forCartesian coordinates: they uniquely identifyeach point along the street, and they encodeproximity: regarding house no. 3, house no.5 is further away than no. 4.

What the coordinates do not allow withoutadditional information is f a direct calculationof distances. To calculated distances, we needinformation about the distances between thehouses – how far away is the mark in front ofno. 1 from that in front of no. 2? How faraway is the no. 2 mark from the no. 3 mark,and so on? If we know that, say, the no. 4 andno. 5 marks are 6.8 m apart, then we knowthat our point 4.5 is 3.4 m from the mark no.4, and the same distance from the no. 5 mark.

With this additional information, we canuse our coordinates to compute distancesbetween points along the street – as long aswe have that additional input, the informa-tion about how far the houses are apart. Thisadditional information is our first example ofa metric: a set of information that allows youto translate coordinate di↵erences into dis-tances. Let us look at a more general, two-dimensional example.

The metric of a general 2-dimensionalsurface

Consider an idealized rocky landscape, withhills and valleys, perfectly smooth and pol-ished, without any breaks or sharp edges.That landscape is the stand-in for a generaltwo-dimensional surface. Imagine that wedraw a two-dimensional Cartesian coordinatesystem onto a su�ciently large rubber sheet.We do not just draw the axes, but all coordin-ate lines: the lines corresponding to x = 1,x = 1.1, x = 1.2 and so on, and correspond-ing lines for y = 0.5, y = 0.51, and y = 0.52and many, many more. In reality, we canonly draw a finite grid of lines, of course; inour thought experiment, we could imagine wehad drawn all the lines. We also label all thelines, so whenever we see a line on that rubber

sheet, we can identify which line x = const.or y = const. we are looking at.

Each point on the rubber sheet has a uniquecoordinate label. After all, that point will besituated on exactly one line x = const. andon one line y = const.; those two lines definethat point’s pair of coordinate values (x, y).

Now imagine that we spread the rubbersheet across our rocky landscape, making surethe sheet covers the surface tightly, with nopockets of air in between, and no wrinkles,each part of the sheet covering a correspond-ing part of the landscape. Evidently, therewill be many places where we will need tostretch the rubber sheet to make sure it fitsthe surface snugly. Our coordinate lines willbecome general, curved lines on the surface.

Even the distorted rubber sheet is su�cientto define a coordinate system on our hilly sur-face. After all, distortion and stretching donot change the fact that every point of thesurface will have one x coordinate line andone y coordinate line intersecting at exactlythat point on the rubber sheet. Every pointof the surface has an (x, y) coordinate pair.

When we take a step back to look at ourrubber-sheet-covered landscape, the coordin-ate lines will in general look wavy and dis-torted. Figure 2 shows an example of whatwe might see, at least for a selected few co-ordinate lines. For the lines shown, you couldimmediately find the point A = (8.3, 5.4) orthe point B = (8.7, 5.7), or any of the nearly80 additional points lying on the intersectionof the visible coordinate lines in figure 2. Ifa finer grid were shown, you could find manymore points; an idealized, infinitesimally finegrid, including all coordinate lines, would al-low you to find coordinates for every point inthe region visible in figure 2 by looking at thecoordinate lines intersecting at that particularpoint.

One glance at figure 2 will show you thatthis is not a Cartesian coordinate system on aplane surface. The wavy lines are a dead give-away. (Note, though, that simply by lookingat the lines in this way, you could not tellwhether this was a distorted rubber sheet ona plane, or on a more complex surface!)

Astronomy from 4 Perspectives 18 Jena 2015

x =8.1

y = 5.1

x =8.2

y = 5.2

x =8.3

y = 5.3

x =8.4

y = 5.4

x =8.5

y = 5.5

x =8.6

y = 5.6

x =8.7

y = 5.7

x =8.8

y = 5.8

x =8.9

y = 5.9

B

A

Figure 2: The rubber sheet coordinate system on our surface, as seen by a birds-eyeobserver, showing selected coordinate lines x = const. and y = const.

4

Fig. 2: The rubber sheet coordinate systemon our surface, as seen by a birds-eyeobserver, showing selected coordinatelines x = const. and y = const.

But what happens when you zoom in on oneparticular point of the surface (and coveringrubber sheet)? There is a more familiar caseof such a zooming-in, namely the infinitesimalstraightness that emerges when you look at aparticular point on a di↵erentiable curve plot-ted in the usual xy coordinate system: zoomin su�ciently, and the portion of the curveyou are looking at will increasingly resemblea straight line. This is the visual motiva-tion for defining the derivative of that func-tion, which is closely related to the line you’dget if you had an infinitely strong zoom, let-ting you see the infinitesimal neighbourhoodof your chosen point.

The properties of our rocky surface(smooth, no breaks or sharp edges) and theprocedure that gave us the coordinate lines(distorting the rubber sheet to fit the surfacesnugly) suggests that these lines, too, are dif-ferentiable. By that criterion, when we zoomin su�ciently close, we should be able to makethe criss-crossing, general/curved coordinatelines look more and more like straight line seg-ments. This suggests that, at suitably highmagnification, the region around each pointshould look as shown in figure 3.

In particular, the straight line segmentsmesh together just as one would expect in

a plane with Euclidean geometry. Our two-dimensional version of infinitesimal straight-ness thus reads: Zoom su�ciently close to asmooth and possibly curved surface, and thesmall (in the limit: infinitesimal) region youare looking at will be indistinguishable froma small subset of a Euclidean plane.

We are using this property of curved sur-faces implicitly whenever we consult a com-mon road-map – which, after all, maps a smallportion of the surface of a sphere, namely theglobe, onto a flat piece of paper.

Let us focus on the area delineated by theparallelogram marked in figure 3. If we ap-proximate the small region in question as asmall subset of a Euclidean plane, then thecoordinate lines on that plane define a co-ordinate system that is almost, but not quiteCartesian: the coordinate lines are indeedstraight lines, but the x and y axis are notorthogonal to each other.

But what happens when you zoom in on one particular point of the surface (andcovering rubber sheet)? There is a more familiar case of such a zooming-in, namelythe infinitesimal straightness that emerges when you look at a particular point on adi↵erentiable curve plotted in the usual xy coordinate system: zoom in su�ciently,and the portion of the curve you are looking at will increasingly resemble a straightline. This is the visual motivation for defining the derivative of that function, which isclosely related to the line you’d get if you had an infinitely strong zoom, letting yousee the infinitesimal neighbourhood of your chosen point.

The properties of our rocky surface (smooth, no breaks or sharp edges) and theprocedure that gave us the coordinate lines (distorting the rubber sheet to fit the sur-face snugly) suggests that these lines, too, are di↵erentiable. By that criterion, when wezoom in su�ciently close, we should be able to make the criss-crossing, general/curvedcoordinate lines look more and more like straight line segments. This suggests that,at suitably high magnification, the region around each point should look as shownin figure 3. In particular, the straight line segments mesh together just as one would

BBBBBBBBBBBBBBBBBBBBBB

Figure 3: A small neighbourhood around the point B = (8.7, 5.7) on the smooth surfacein figure 2, su�ciently magnified for the coordinate lines to look like segments ofstraight lines. Next, we explore the geometry of the region inside the parallelogram.

expect in a plane with Euclidean geometry. Our two-dimensional version of infinites-imal straightness thus reads: Zoom su�ciently close to a smooth and possibly curvedsurface, and the small (in the limit: infinitesimal) region you are looking at will beindistinguishable from a small subset of a Euclidean plane.

We are using this property of curved surfaces implicitly whenever we consult acommon road-map – which, after all, maps a small portion of the surface of a sphere,namely the globe, onto a flat piece of paper.

Let us focus on the area delineated by the parallelogram marked in figure 3. If weapproximate the small region in question as a small subset of a Euclidean plane, thenthe coordinate lines on that plane define a coordinate system that is almost, but notquite Cartesian: the coordinate lines are indeed straight lines, but the x and y axis arenot orthogonal to each other.

5

Fig. 3: A small neighbourhood around thepoint B = (8.7, 5.7) on the smooth sur-face in figure 2, su�ciently magnifiedfor the coordinate lines to look like seg-ments of straight lines.

This need not keep us from calculating thedistance of any point P in the parallelogramfrom the basepoint B = (8.7, 5.7), though.The geometry of the situation can be seenin figure 4. More generally, let us place ourbasepoint at B = (x, y). What do we need toknow in order to determine distances? First

Astronomy from 4 Perspectives 19 Jena 2015

of all, just as in the example with the housenumbers, we need a scale factor. Consider asmall shift dx in the x coordinate, moving usfrom P1 = (x, y) to P2 = (x + dx, y). Definethe scale factor a linking the coordinate dis-tance dx and the distance P1P2 by

P1P2 = a dx. (3)

Analogously, we define a scale factor b forsmall shifts in the y direction. At B, theangle between the x and y coordinate line is↵. From the law of cosines, it follows that fora general coordinate shift (dx, dy), the dis-tance ds between B = (x, y) and the pointP = (x + dx, y + dy) is given by

ds2 = a2 dx2 + 2 ab cos(↵) dx dy + b2 dy2

⌘ g11 dx2 + 2g12 dx dy + g22 dy2, (4)

where the expression on the right serves todefine the metric coe�cients g11, g12, and g22.Such an expression linking infinitesimal dis-tances ds and coordinate shifts dx, dy, . . . iscalled a line element.

Just as in Pythagoras’ theorem, this expres-sion is second order in the coordinate shifts.The information linking coordinate shifts anddistances is encoded in the metric coe�cients.Taken together, the coe�cients form the met-ric at that particular point. At other loca-tions on the surface, there is an analogousformula linking coordinate shifts and infinites-imal distances; in general, the metric coe�-cients g11, g12, and g22 will take on other val-ues than at our original basepoint B as wemove to di↵erent locations. In other words:the metric coe�cients are functions of posi-tion, g11(x, y), g12(x, y), and g22(x, y), and sois the metric.

Once we know these functions, we can an-swer all questions about the inner geometryon the surface. In short, the inner geometrydeals with all questions that a hypotheticaltwo-dimensional being that is living on thesurface can ask about curves in the surface,the geometric objects that can be constructedfrom such curves, and the various distancesand angles involved. The answers, arrived atusing the metric, could each be checked by ap-propriate measurements taken on the surface.

On the other hand, the metric cannot tell us

anything about the exterior geometry, that is,the specifics of how the two-dimensional sur-face is embedded in three-dimensional space.For a simple example, imagine all the waysone can embed a two-dimensional sheet of pa-per in space; the geometry of triangles etc.on the sheet remains Euclidean even whenthe embedding changes. In fact, formulatinggeometry in terms of a metric provides themeans to discuss the geometry of curved sur-faces without the need for such embeddings!

The metric coe�cients introduced here arethe central elements of the general descrip-tion for the inner geometry of smooth sur-faces, which was found by Carl Friedrich Gauß(1777–1855). Bernhard Riemann (1826–1866)generalized this description to spaces of arbit-rary dimension.

Figure 3: A small neighbourhood around the point (8.7, 5.7) on the smooth surface infigure 2, su�ciently magnified for the coordinate lines to look like segments of straightlines. The parallelogram marks the region whose geometry we will explore in the nextfigure.

We are using this property of curved surfaces implicitly whenever we consult acommon road-map – which, after all, maps a small portion of the surface of a sphere,namely the globe, onto a flat piece of paper.

Let us focus on the area delineated by the parallelogram marked in figure 3. If weapproximate the small region in question as a small subset of a Euclidean plane, thenthe coordinate lines on that plane define a coordinate system that is almost, but notquite Cartesian: the coordinate lines are indeed straight lines, but the x and y axis arenot orthogonal to each other.

This doesn’t keep us from calculating the distance of any point P in the parallel-ogram from the basepoint B = (8.7, 5.7), though. The geometry of the situation canbe seen in figure 4. More generally, let us place our basepoint at B = (x, y). What

P

B

dxdy

↵

Figure 4: Parallelogram adjoining the basepoint B = (8.7, 5.7). What is the distancebetween B and some point P whose coordinates are shifted against those of B by dx inthe x direction, dy in the y direction?

5

Fig. 4: Parallelogram adjoining the basepointB. What is the distance between Band some point P whose coordinatesare shifted against those of B by dx inthe x direction, dy in the y direction?

Spacetime metric

The properties of special relativity can bewritten using a metric, as well. At first glance,the line element looks a bit weird. While it iswritten as a square, the line element can be-come negative! In the usual coordinates of aninertial system S, using Cartesian coordinatesfor space, it can be written as

ds2 = �c2 dt2 + dx2 + dy2 + dz2, (5)

where c is the speed of light in vacuum. Allthe usual special-relativistic e↵ects, such astime dilation and length contraction, can bederived from this metric, which is known as

Astronomy from 4 Perspectives 20 Jena 2015

the Minkowski metric.

The starting point for general relativityis a generalization of the metric (5). Themost general form of the metric will containall second-order products of coordinate di↵er-ences (such as dx·dt or dx·dz, among others),and the metric coe�cients can be functions ofall the four coordinates.

For our simple analysis of cosmological andgravitational wave spacetimes, two generalproperties of such general spacetime metricsare su�cient: If we confine ourselves to dt =0, then the remaining non-zero part of themetric can be used in direct analogy to whatwe did with the metric of our two dimensionalsurface to relate spatial coordinate shifts tospatial distances.1

The second general property is that thetrajectories x(t), y(t), z(t) taken by light al-ways satisfy ds2 = 0, more specifically: ifdx, dy, dz are the shifts in the spatial coordin-ate of a photon as the coordinate time intervaldt passes, the line element ds2 for this particu-lar set of coordinate shifts dx, dy, dz, dt mustvanish. It is straightforward to see that, inthe case of (5) in special relativity, this cor-responds to light moving with the usual speedof light c.2

In general relativity, the geometry of space-time (encoded by the functions that specifythe metric coe�cients) is determined by theEinstein Field Equations (EFE). The EFEare second order di↵erential equations linkingthe first and second derivatives of the metriccoe�cients with the mass, energy, momentumand pressure of whatever matter is present inthe spacetime in question, which serves as asource of gravity. 3 A set of matching metriccoe�cients and suitable source terms, linkedby the EFEs, is called a solution of the EFEs.All models of physical situations in the frame-work of general relativity are formulated interms of suitable solutions.

Using the EFEs, it is possible to calculate ina systematic manner the deviations from clas-sical Newtonian gravity which occur in situ-

ations with comparatively weak gravity. New-tonian gravity itself can be described, in a co-ordinate system close to that chosen in clas-sical physics, as a location-dependent coe�-cient g00 in front of the metric’s dt2 term.

Metrics for an expanding cosmos and forgravitational waves

The simplest metric for a homogeneous andisotropic expanding universe, and the met-ric on which current cosmological models arebased, is

ds2 = �c2 dt2+a(t)2⇥dx2 + dy2 + dz2

⇤. (6)

In general relativity, as more generally in sci-entific modelling, choosing suitable coordin-ates is very important. In the case of (6),the coordinates have been chosen adapted tothe situation as follows: consider idealizedgalaxies that follow cosmic expansion withoutany additional velocity components (“pecu-liar velocity” due to motion within galaxyclusters). Such galaxies are said to move withthe Hubble flow; in our coordinate system,they are assigned constant coordinate valuesx, y, z. (This is known as using comoving co-ordinates – the coordinate system moves alongwith the galaxies.) The time coordinate t ismeasured by clocks moving alongside galax-ies in the Hubble flow. These clocks are syn-chronized in exactly the right way that, forany fixed time t = const., the local averagedensity of the universe is the same at eachlocation. (In other words: our notion of sim-ultaneity is adapted to the homogeneity of theuniverse.)

This is the simplest case of Friedmann-Lemaıtre-Robertson-Walker universe (FLRWuniverse), namely an expanding universe withEuclidean spatial geometry (in the lingo ofFLRW solutions, k = 0; the other possibilitiesfor FLRW universes are hyperbolic geometryk = �1 and spherical geometry k = +1).

The function a(t) is called the cosmic scalefactor. Its role can be read o↵ directly from

1There is a caveat when it comes to interpreting these spatial distances; in general, the interpretation dependson the meaning of the time coordinate that was chosen – in line with the fact that already plays an importantrole in special relativity, namely that measuring distances depends on one’s notion of simultaneity.

2For the easiest way to see this, restrict yourself to light propagation in the x direction only. Then ds2 = 0translates to |dx/dt| = c.

3An elementary treatment can be found in [1]

Astronomy from 4 Perspectives 21 Jena 2015

the metric (6): Assume that, at a fixed mo-ment in time t, we determine the distancebetween two galaxies A and B. Without lossof generality we can assume the y and z co-ordinates of these two galaxies to be equal(since we can always rotate our coordinatesystem so that two given galaxies A and Bare separated in the x direction only). Weobtain the distance between A and B by in-tegrating the (infinitesimally) small distanceelement ds along the straight line joining Aand B, namely

dAB

(t) =

BZ

A

ds = a(t)

xBZ

xA

|dx|

= a(t) · |xB

� xA

|. (7)

Evidently, all distances between arbitrarypairs of galaxies in the Hubble flow changeover time are proportional to the cosmic scalefactor. For any two galaxies A and B, andany two cosmic times t1 and t2, we have

dAB

(t1) =a(t1)

a(t2)dAB

(t2). (8)

This is the formula encoding what is meant bycosmic expansion: distances between distantgalaxies changing proportionally to the samecosmic scale factor.

Next, we turn to gravitational waves:minute, propagating disturbances of space-time geometry, travelling at the speed of light.Gravitational waves one can hope to detectare typically produced by the fast, acceleratedmovement of compact objects such as blackholes or neutron stars.

The simplest metric for a gravitational wavein empty space can be written as

ds2 = �c2 dt2 + [1 + h(t � z/c)]dx2

+[1 � h(t � z/c)]dy2 + dz2. (9)

In the simplest case, the gravitational wavestrain is a sine function

h(t) = A · sin(!t), (10)

possibly with an extra phase shift that is notincluded here, where the amplitude A is ex-tremely small for realistic gravitational wavesignals, A ⇠ 10�21. Similar to the cosmolo-

gical case, the coordinates have been chosento ensure each set of fixed spatial coordinates(x, y, z) can be associated with a particle infree fall, which will keep its position in thecoordinate systems even when a gravitationalwave passes. (In other words, trajectorieswith x = const., y = const. z = const. aregeodesics, that is: possible trajectories for un-constrained particles in free fall.) In the ab-sence of a gravitational wave, h = 0, the co-ordinates reduce to the standard coordinatesof special relativity.

The gravitational wave is propagating atthe speed of light c; in the metric (9), propaga-tion is in the positive z direction. The metricis a special case of a plane-fronted wave withparallel propagation, “pp wave” for short, de-rived from a linear approximation that treatsgravitational waves as small deviations froman otherwise flat spacetime. The specific formgiven in (9) corresponds to the transversaltraceless gauge, “TT gauge” for short, andcan be found in many university-level textbooks on general relativity.

The action of the gravitational wave (9) canbe seen most readily by examining a group ofparticles floating freely in the xy plane. Fig-ure 5 shows such a group of free particles,arranged to form a circle, at di↵erent phasevalues for the gravitational wave. The pagecorresponds to the xy plane, with the grav-itational wave propagating from the back to-wards the observer. The pattern of stretch-ing and shrinking, with maximal stretching inthe horizontal direction while there is max-imal shrinking in the vertical direction, andvice versa, is characteristic for gravitationalwaves, a direct consequence of what is meantwhen the waves are called quadrupole distor-tions. The direction of motion is perpendicu-lar to the distortions, in other words: gravit-ational waves are transversal.

Similarities and di↵erences

Some similarities between the two metrics(6) and (9) are immediately obvious. Inboth cases, the deviations from flat spacetime(Minkowski metric) is in the spatial part only.In both cases, there are time-dependent met-ric coe�cients in front of the terms dx2 anddy2 that do not depend on x and y.

Astronomy from 4 Perspectives 22 Jena 2015

� = 0�

� = 45�

� = 90�

� = 135�

� = 180�

� = 225�

� = 270�

� = 315�

Figure 5: Gravitational wave acting on a circle of freely floating particles. Plotted arethe distances relative to a freely floating particle in the center of each image for variousphases of the gravitational wave. The wave is propagating at right angles to the imageplane, e.g. directly towards the viewer. The amplitude has been exaggerated by manyorders of magnitude to render the e↵ects of the wave visible.

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Fig. 5: Gravitational wave acting on a circle of freely floating particles. Plotted are the distancesrelative to a freely floating particle in the center of each image for various phases of thegravitational wave. The wave is propagating at right angles to the image plane, e.g.directly towards the viewer. The amplitude has been exaggerated by many orders ofmagnitude to render the e↵ects of the wave visible.

Since the only term involving dt is, in bothcases, �c2dt2, the time coordinate is the timeas shown by blocks that are at rest at constantcoordinate locations. Also in both cases, con-stant coordinate locations are a form of freefall, in other words: In the absence of externalnon-gravitational forces, particles at rest inthe spatial coordinates, floating in space, willremain floating at those same coordinate val-ues of their own accord.

On the other hand, there is a fundamentaldi↵erence in that in cosmic expansion, alldirections of space are on an equal footing,whereas a gravitational wave only a↵ects thetwo spatial directions orthogonal to its direc-tion of propagation.

In the xy plane, at constant z and withdz = 0, we can write both metrics in the form

ds2 = �c2 dt2 + ax

(t)2dx2 + ay

(t)2dy2. (11)

From this generalized metric, we can derivesome common e↵ects that are present bothin both spacetimes: in an expanding universeand as a gravitational wave passes by.

Frequency shift for light

Consider light propagating in the xy planewith the general metric (11). More spe-cifically, let us confine our attention to lightpropagating in one of the coordinate direc-tions only, say: the x direction. Using sym-

Astronomy from 4 Perspectives 23 Jena 2015

metry considerations, obvious for the cosmo-logical case and somewhat more subtle in thecase of the gravitational wave, it can be shownthat this is no loss of generality.

The part of our metric dealing with the xand t directions only is

ds2 = �c2 dt2 + ax

(t)2dx2. (12)

Let x(t) describe the propagation of lightalong the x direction. Per definition, lightpropagation means ds2 = 0, and we obtain

cdt

ax

(t)= ±dx. (13)

The two di↵erent signs correspond to the twopossible propagation directions of light, in thepositive or negative x direction.

Integrating up, we find that for light emit-ted at the location x

e

at the time te

and re-ceived at the location x

r

< xe

at the time tr

,we have

c

trZ

te

dt

ax

(t)= x

e

� xr

. (14)

Independent of the specific form of ax

(t), wecan derive the following: Consider two pulsesof light emitted at the same location x

e

attimes t

e

and te

+ �te

, which arrive at the loc-ation x

r

< xe

at times tr

and tr

+ �tr

. Ac-cording to (14), we have

0 =

tr+�trZ

te+�te

dt

ax

(t)�

trZ

te

dt

ax

(t)

=

2

4tr+�trZ

tr

+

trZ

te

�te+�teZ

te

�trZ

te

3

5 dt

ax

(t)

=

2

4tr+�trZ

tr

�te+�teZ

te

3

5 dt

ax

(t)

⇡ �te

ax

(te

)� �t

r

ax

(tr

). (15)

The argument remains valid when we considernot light pulses, but consecutive wave crests oflight waves. In this case, the �

t

correspond tothe oscillation period of the light, and are thusproportional to the light wave’s wavelength �.

Thus, (15) entails

�r

�e

=ax

(tr

)

ax

(te

). (16)

In an expanding universe, the scale factor hasgrown between the time t

a

and the later timetr

, so ax

(tr

) > ax

(te

). Thus, for distant galax-ies whose light reaches us at the present time,the e↵ect derived here corresponds to a sys-tematic redshift known as the cosmic redshift.

On the other hand, light propagating atright angles to the direction of a gravitationalwave is subject to a series of periodic red- andblue shifts that arise from the metric in ex-actly the same manner as the cosmic redshift.

Bound systems

Next, consider a bound system in a spacetimedescribed by the metric (11). This is morecomplicated than simply tracing the move-ment of particles in free fall (which amountsto finding the spacetime’s straightest possiblelines, or geodesics), as we need to take into ac-count both gravitational acceleration and thenon-gravitational forces responsible for keep-ing the system bound (or not).

The most straightforward way of obtainingat least an approximate description of whathappens to a bound system is based on Ein-stein’s principle of equivalence, one of the fun-damental principles of general relativity. Theequivalence principle is the spacetime ana-logue of our zooming-in on an infinitesimal re-gion of our curved surface discussed on page17 and the following. In that case, a su�-ciently high zoom factor made the region un-der scrutiny look indistinguishable from a sub-set of a Euclidean space. In fact, by a changeof coordinates, we could have replaced thenon-orthogonal coordinate system in figure 4by a proper orthogonal Cartesian system —at least locally. The equivalence principle ap-plies the same zoom-in principle to a generalspacetime: At least locally, in the infinitesimalneighbourhood of any event, spacetime is in-distinguishable from the flat spacetime of spe-cial relativity. By a suitable coordinate trans-formation, we can make that infinitesimal re-gion look like flat spacetime in the usual co-ordinates, distances and light propagation de-

Astronomy from 4 Perspectives 24 Jena 2015

scribed by the Minkowski metric (5). Thereis a simple physical interpretation for the ori-gin of such a locally Minkowskian system: theorigin is the location of an observer in free fall.

Let us choose just such a system; in fact, wecan keep the origin to be the same as in ourearlier version of the metric (11), given thatthe origin and any other fixed coordinate loca-tion correspond to the trajectories of particlesin free fall. In this system, we will consider thenon-relativistic limit (applicable to particleswhose velocities are slow compared to c) andexamine the consequences for a bound systemdescribed with the help of Newtonian phys-ics.4

For simplicity, we will again focus on onedirection of space only, choosing once morethe x direction, and the restricted metric (12).One simple change is su�cient to make thismetric look like that of special relativity atleast locally: we introduce the new spatial co-ordinate x = a

x

(t) · x, whose coordinate dif-ferences amount to spatial distances along thex axis, just as in the usual classical Cartesiancoordinate system. Direct calculation showsthat, for a free particle with x = const., wehave

¨x =ax

(t)

ax

(t)· x. (17)

Through the lens of classical physics, this isan inertial acceleration a↵ecting all particlesequally. In systems with this kind of inertialacceleration, Newton’s second law of mechan-ics takes on the modified form

m

¨x � a

x

(t)

ax

(t)· x

�= F

x

. (18)

In the case of a gravitational wave with theax

(t) defined in (9), and neglecting higher-order terms in A, this equation can be writtenas

m

¨x +

A!2

2cos(![t � z/c]) · x

�= F

x

. (19)

The simplest case is for the non-gravitationalforce F

x

to follow Hooke’s law. The result

is the simplest model for a so-called resonantgravitational wave detector: an oscillator witha forced sinusoidal oscillation. The detectorsbuilt from the 1960s onwards typically weremetal cylinders with a length on the scale ofa few meters, a meter in diameter, and withhigh quality factors to ensure that an oscilla-tion excited by a passing gravitational wavefollowing (19) would have as large an amp-litude as possible. These detectors were usedin the first (and unsuccessful) attempts to dir-ectly detect gravitational waves.

The same formula can be applied to studythe e↵ects of cosmic expansion on a boundsystem. Instead of the more general form ofthe cosmic scale factor a(t), we consider aTaylor expansion,

a(t) = a0 + H0(t � t0)

+1

2↵ (t � t0)

2 + O(3) (20)

where H0 is known as the Hubble constant.For a bound system with Coulomb-like cent-ral force

Fc

= �mC

r2, (21)

and with the realization that our formula (18)doesn’t just apply to x, but equally to a ra-dial coordinate r measuring the distance fromthe origin, we have all the tools to model oursystem. As usual, our model includes a termdepending on the angular momentum L = 'to make for an e↵ective potential5

¨r = ↵ · r � C

r2+

L2

R3. (22)

Such a system will remain bound as long as

r < rc

=

✓C

↵

◆1/3

(23)

with a critical radius rc

. Using modern val-ues for the cosmological parameters, namelythe Hubble constant H0 = 68 km/(s · Mpc) =2.2 ·10�18 s�1 and the deceleration parameter

4Both this simplified pseudo-Newtonian derivation and a more rigorous analysis can be found in [2] as well asin [3]

5Cf. [3]6Based on measurements of the ESA satellite Planck and gravitational lensing data as per Planck Collaboration[4]

Astronomy from 4 Perspectives 25 Jena 2015

q0 = �0.54,6 we obtain

↵ = 2.66 · 10�36 s�2. (24)

For a Coulomb system consisting of a protonand an electron,

C = e2/(4⇡"0me

) = 253.27 m3/s2, (25)

which corresponds to a critical radius rc

=30,54 AU. Hence, a hydrogen atom is stableagainst the influence of (accelerated) cosmicexpansion as long as the electron is less dis-tant from its proton than Neptune is from theSun!

For the gravitational attraction acting on aplanet that orbits the Sun, we have

C = GM� ⇠ 1020 m3/s2, (26)

corresponding to a critical radius of rc

= 390light-years. A planetary orbit around the Sunis stable against the current (accelerated) cos-mic expansion as long as it is smaller than thatvery large radius.

One key point concerning these formulae isthat the only influence of cosmic expansion onbound systems is a. This shows quite clearlywhere interpretations of cosmic expansion asan “expansion of space” or, even worse, as“new space being created in the course of theexpansion” fall short. Instead, the situation issimilar to the generic dynamical situation inphysics: dynamical e↵ects do not arise fromfirst derivatives (in the case of cosmic expan-sion: the Hubble constant), but from accelera-tion terms. Objects are not “carried along” bycosmic expansion; they experience the sameacceleration acting on objects in the Hubbleflow.

Interferometric gravitational wavedetectors

With what we have learned in the precedingsections, we can also understand the basicsof interferometric gravitational wave detectorssuch as those used for the first direct detec-tion of gravitational waves in mid-September2015, one-and-a-half weeks after our gravita-tional wave summer school in Jena.

8 Interferometric gravitational wave detectorsWith what we have learned in the preceding sections, we can also understand the basicsof interferometric gravitational wave detectors such as those used for the first directdetection of gravitational waves in mid-September 2015, one-and-a-half weeks afterour gravitational wave summer school in Jena.

L

M2

M1

PD

B

Figure 6: Simple Michelson interferometer. Interferometric gravitational wave detec-tors are more complicated versions of this.

The basic structure of such a detector is that of a Michelson interferometer, thesimplest version of which is shown in figure 6: Light from a laser source L propagatesto the beam splitter B. Half of the light continues on to the mirror M1, while theother is reflected towards the mirror M2. After reflection at the respective mirrors, thelight returns to the beam splitter B, where half of each portion moves on towards thephotodetector PD.

The mirrors M1 and M2 and the beam splitter B are suspended as multiple pendu-lums to isolate them as far as possible from external disturbances. When it comes totheir reactions to a passing gravitational wave, and their motion back and forth alongthe directions of M1–B and M2–B, respectively, this makes them act approximatelylike particles in free fall. Interferometric detectors are commonly adjusted so that inthe absence of a gravitational wave, hardly any light escapes towards the photodetec-tor PD; in other words: there is almost complete destructive interference of the lightmoving from M1 and M2 into the direction of PD.

A gravitational wave passing through such a detector – in the simplest case: or-thogonal to the image plane of figure 6 – makes a twofold di↵erence. For one, the

15

Fig. 6: Simple Michelson interferometer. In-terferometric gravitational wave de-tectors are more complicated versionsof this.

The basic structure of such a detectoris that of a Michelson interferometer, thesimplest version of which is shown in figure6: Light from a laser source L propagates tothe beam splitter B. Half of the light continueson to the mirror M1, while the other is reflec-ted towards the mirror M2. After reflectionat the respective mirrors, the light returns tothe beam splitter B, where half of each por-tion moves on towards the photodetector PD.

The mirrors M1 and M2 and the beam split-ter B are suspended as multiple pendulumsto isolate them as far as possible from ex-ternal disturbances. When it comes to theirreactions to a passing gravitational wave, andtheir motion back and forth along the direc-tions of M1–B and M2–B, respectively, thismakes them act approximately like particlesin free fall. Interferometric detectors arecommonly adjusted so that in the absenceof a gravitational wave, hardly any light es-capes towards the photodetector PD; in otherwords: there is almost complete destructiveinterference of the light moving from M1 andM2 in the direction of PD.

A gravitational wave passing through such adetector – in the simplest case: orthogonal tothe image plane of figure 6 – makes a twofolddi↵erence. For one, the distances between the

Astronomy from 4 Perspectives 26 Jena 2015

beam splitter and the end mirrors will changein the same way as the distances betweenthe freely falling particles in figure 5; in thesimplest case, one arm will be stretched whileat the same time the other arm will be shrunk.Destructive interference cannot be maintainedunder such conditions, and light will leak outtowards the photodetector. After all, the timeit takes for wave crests and troughs via M1and M2 to the detector PD will change as therelative armlengths change.

An additional e↵ect are the frequency shiftsfor light, discussed at page 22 and the fol-lowing. As arms are stretched and shrunk,light within the detector is red- and blueshif-ted accordingly, proportional to the relevantscale factor. To light waves that have dif-ferent wave lengths can never have completedestructive interference; this e↵ect contrib-utes to the light leaking out at the photo-detector, as well. When the time light re-mains inside the detector is short comparedto the oscillation period of the gravitationalwave, as in current ground-based detectors,the consequences of this e↵ect are much smal-ler than the change in armlengths. For muchlonger light-travel times, such as in the di↵er-ent varieties of the proposed space-based de-tector LISA, gravitational-wave-induced fre-quency shifts become more important.7

Conclusions

The general-relativistic descriptions of cos-mic expansion and of gravitational waves havesimilarities that are helpful in understandingboth phenomena. In their simplest incarna-tions, both feature a scale factor or multiplescale factors in front of an otherwise flat spa-

tial metric. In cosmology, this scale factorgoverns the changing distances of particles inthe Hubble flow, in the case of gravitationalwaves the changing distances between variouscomponents of an interferometric detector.

When non-gravitational forces are present,it is easiest to change to a pseudo-Newtonianpicture. Here, the second derivatives of thescale factors cause inertial accelerations thatcan be contrasted with the acceleration aparticle experiences through a Coulomb-likeforce. In the cosmological case, this helps todistinguish kinematic and dynamic compon-ents, showing that particles are not “whiskedalong” with the Hubble flow, but given ar-bitrary initial conditions, react only to thesecond-order, dynamical influence of scale-factor expansion. In the case of gravitationalwaves, the corresponding description allowsfor an understanding of resonant wave detect-ors.

A spacetime metric directly governs thepropagation of light, and the scale-factor met-rics studied here allow for simple calculationsof the wavelength shifts experienced by light.In the cosmological case, this yields the fam-ous cosmological redshift. For gravitationalwave, it yields red- and blueshifts that becomeimportant for future large-scale, space-baseddetectors.

Acknowledgements

I am grateful to B. Schutz for discussions dur-ing the summer school that proved helpful forthis text and corresponding lecture, and to T.Muller for comments on an earlier draft of thismanuscript.

References

[1] J. C. Baez & E. F. Bunn: The meaning of Einstein’s equation, American Journal of Physics73 (2005), pp. 644–652. DOI:10.1119/1.1852541

[2] D. Giulini: Does cosmological expansion a↵ect local physics, Studies in History and Philo-sophy of Modern Physics 46A (2014), pp. 24-37; DOI: 10.1016/j.shpsb.2013.09.009

[3] M. Carrera & D. Giulini: Influence of global cosmological expansion on local dynamicsand kinematics,Reviews of Modern Physics 82 (2008), pp. 169-208, DOI: 10.1103/RevMod-Phys.82.169

7Cf. [5]

Astronomy from 4 Perspectives 27 Jena 2015

[4] Planck Collaboration: Planck 2013 results. XVI. Cosmological parameters, Astronomy &Astrophysics 571 (2014), article id. A16. DOI:10.1051/0004-6361/201321591

[5] P. Saulson: If light waves are stretched by gravitational waves, how can we use light as aruler to detect gravitational waves?, American Journal of Physics, Volume 65, Issue 6, pp.501-505 (1997). DOI: 10.1119/1.18578