EspacioTiempo - A Einstein

7/29/2019 EspacioTiempo - A Einstein

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The revolution of modern physics began to bereflected in the Encyclopdia Britannica in theTwelfth Edition (1922), with James (later Sir

James) H. Jeans' article on "Relativity." In theThirteenth Edition (1926) a wholly new topic,"Space-Time," was discussed by the person mostqualified in all the world to do so, Albert Einstein.The article is challenging but rewarding.

SPACE-TIME All our thoughts and concepts are calledup by sense-experiences and have a meaning only inreference to these sense-experiences. On the otherhand, however, they are products of the spontaneousactivity of our minds; they are thus in no wise logicalconsequences of the contents of these sense-experiences. If, therefore, we wish to grasp the essenceof a complex of abstract notions we must for the onepart investigate the mutual relationships between theconcepts and the assertions made about them; for theother, we must investigate how they are related to theexperiences.

So far as the way is concerned in which concepts areconnected with one another and with the experiencesthere is no difference of principle between the concept-systems of science and those of daily life. The concept-systems of science have grown out of those of daily lifeand have been modified and completed according to theobjects and purposes of the science in question.

The more universal a concept is the more frequently itenters into our thinking; and the more indirect itsrelation to sense-experience, the more difficult it is forus to comprehend its meaning; this is particularly thecase with pre-scientific concepts that we have beenaccustomed to use since childhood. Consider theconcepts referred to in the words "where," "when,""why," "being," to the elucidation of which innumerablevolumes of philosophy have been devoted. We fare nobetter in our speculations than a fish which should striveto become clear as to what is water.

SPACE

In the present article we are concerned with the


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meaning of "where," that is,of space. It appears thatthere is no quality containedin our individual primitivesense-experiences that maybe designated as spatial.Rather, what is spatialappears to be a sort of orderof the material objects of experience. The concept"material object" musttherefore be available if concepts concerning spaceare to be possible. It is thelogically primary concept.

This is easily seen if weanalyse the spatial concepts for example, "next to,""touch," and so forth, that is, if we strive to becomeaware of their equivalents in experience. The concept"object" is a means of taking into account thepersistence in time or the continuity, respectively, of certain groups of experience-complexes. The existenceof objects is thus of a conceptual nature, and themeaning of the concepts of objects depends wholly ontheir being connected (intuitively) with groups of elementary sense-experiences. This connection is thebasis of the illusion which makes primitive experienceappear to inform us directly about the relation of material bodies (which exist, after all, only in so far asthey are thought).

In the sense thus indicated we have (the indirect)experience of the contact of two bodies. We need do nomore than call attention to this, as we gain nothing forour present purpose by singling out the individualexperiences to which this assertion alludes. Many bodiescan be brought into permanent contact with one anotherin manifold ways. We speak in this sense of the position-relationships of bodies ( Lagenbeziehungen ). The generallaws of such position-relationships are essentially theconcern of geometry. This holds, at least, if we do notwish to restrict ourselves to regarding the propositionsthat occur in this branch of knowledge merely asrelationships between empty words that have been set

Albert EinsteinLeo Baeck Institute / Archive Photos


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up according to certain principles.

Pre-scientific Thought. --Now, what is the meaning of theconcept "space" which we also encounter in pre-scientific thought? The concept of space in pre-scientificthought is characterised by the sentence: "we can thinkaway things but not the space which they occupy." It isas if, without having had experience of any sort, we hada concept, nay even a presentation, of space and as if we ordered our sense-experiences with the help of thisconcept, present a priori . On the other hand, spaceappears as a physical reality, as a thing which existsindependently of our thought, like material objects.Under the influence of this view of space thefundamental concepts of geometry: the point, thestraight line, the plane, were even regarded as having aself-evident character. The fundamental principles thatdeal with these configurations were regarded as beingnecessarily valid and as having at the same time anobjective content. No scruples were felt about ascribingan objective meaning to such statements as "threeempirically given bodies (practically infinitely small) lieon one straight line," without demanding a physicaldefinition for such an assertion. This blind faith inevidence and in the immediately real meaning of theconcepts and propositions of geometry becameuncertain only after non-Euclidean geometry had beenintroduced.

Reference to the Earth. --If we start from the view that allspatial concepts are related to contact-experiences of solid bodies, it is easy to understand how the concept"space" originated, namely, how a thing independent of bodies and yet embodying their position-possibilities(Lagerungsmglichkeiten ) was posited. If we have asystem of bodies in contact and at rest relatively to oneanother, some can be replaced by others. This propertyof allowing substitution is interpreted as "availablespace." Space denotes the property in virtue of whichrigid bodies can occupy different positions. The view thatspace is something with a unity of its own is perhapsdue to the circumstance that in pre-scientific thought allpositions of bodies were referred to one body (referencebody), namely the earth. In scientific thought the earth


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is represented by the co-ordinate system. The assertionthat it would be possible to place an unlimited number of bodies next to one another denotes that space is infinite.In pre-scientific thought the concepts "space" and "time"and "body of reference" are scarcely differentiated at all.A place or point in space is always taken to mean amaterial point on a body of reference.

Euclidean Geometry. --If we consider Euclidean geometrywe clearly discern that it refers to the laws regulatingthe positions of rigid bodies. It turns to account theingenious thought of tracing back all relationsconcerning bodies and their relative positions to the verysimple concept "distance" ( Strecke ). Distance denotes arigid body on which two material points (marks) havebeen specified. The concept of the equality of distances(and angles) refers to experiments involvingcoincidences; the same remarks apply to the theoremson congruence. Now, Euclidean geometry, in the form inwhich it has been handed down to us from Euclid, usesthe fundamental concepts "straight line" and "plane"which do not appear to correspond, or at any rate, notso directly, with experiences concerning the position of rigid bodies. On this it must be remarked that theconcept of the straight line may be reduced to that of the distance. Moreover, geometricians were lessconcerned with bringing out the relation of theirfundamental concepts to experience than with deducinglogically the geometrical propositions from a few axiomsenunciated at the outset.

A hint of this is contained in the theorem: "the straightline is the shortest connection between two points." Thistheorem served well as a definition of the straight line,although the definition played no part in the logicaltexture of the deductions.

Let us outline briefly how perhaps the basis of Euclideangeometry may be gained from the concept of distance.

We start from the equality of distances (axiom of theequality of distances). Suppose that of two unequaldistances one is always greater than the other. Thesame axioms are to hold for the inequality of distances


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as hold for the inequality of numbers.

Three distances AB , BC , CA may, if CA be suitablychosen, have their marks BB , CC , AA superposed onone another in such a way that a triangle ABC results.

The distance CA has an upper limit for which thisconstruction is still just possible. The points A, (BB') andC then lie in a "straight line" (definition). This leads tothe concepts: producing a distance by an amount equalto itself; dividing a distance into equal parts; expressinga distance in terms of a number by means of ameasuring-rod (definition of the space-interval betweentwo points).

When the concept of the interval between two points orthe length of a distance has been gained in this way werequire only the following axiom (Pythagoras' theorem)in order to arrive at Euclidean geometry analytically.

To every point of space (body of reference) threenumbers (co-ordinates) x, y, z may be assigned--andconversely--in such a way that for each pair of points A(x , y , z ) and B (x , y , z ) the theorem holds:

measure-number

AB = {sqroot (x - x ) + (y - y ) + (z - z ) }

All further concepts and propositions of Euclideangeometry can then be built up purely logically on thisbasis, in particular also the propositions about thestraight line and the plane.

These remarks are not, of course, intended to replacethe strictly axiomatic construction of Euclideangeometry. We merely wish to indicate plausibly how all

conceptions of geometry may be traced back to that of distance. We might equally well have epitomised thewhole basis of Euclidean geometry in the last theoremabove. The relation to the foundations of experiencewould then be furnished by means of a supplementarytheorem.

The co-ordinate may and must be chosen so that two


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pairs of points separated by equal intervals, ascalculated by the help of Pythagoras' theorem, may bemade to coincide with one and the same suitably chosendistance (on a solid).

The concepts and propositions of Euclidean geometrymay be derived from Pythagoras' proposition without theintroduction of rigid bodies; but these concepts andpropositions would not then have contents that could betested. They are not "true" propositions but onlylogically correct propositions of purely formal content.

Difficulties. --A serious difficulty is encountered in theabove represented interpretation of geometry in that therigid body of experience does not correspond exactly with the geometrical body. In stating this I am thinkingless of the fact that there are no absolutely definitemarks than that temperature, pressure and othercircumstances modify the laws relating to position. It isalso to be recollected that the structural constituents of matter (such as atom and electron, q.v. ) assumed byphysics are not in principle commensurate with rigidbodies, but that nevertheless the concepts of geometryare applied to them and to their parts. For this reasonconsistent thinkers have been disinclined to allow realcontents of facts ( reale Tatsachenbestnde ) tocorrespond to geometry alone. They considered itpreferable to allow the content of experience(Erfahrungsbestnde ) to correspond to geometry andphysics conjointly.

This view is certainly less open to attack than the onerepresented above; as opposed to the atomic theory it isthe only one that can be consistently carried through.Nevertheless, in the opinion of the author it would not beadvisable to give up the first view, from which geometryderives its origin. This connection is essentially foundedon the belief that the ideal rigid body is an abstractionthat is well rooted in the laws of nature.

Foundations of Geometry. --We come now to thequestion: what is a priori certain or necessary,respectively in geometry (doctrine of space) or itsfoundations? Formerly we thought everything--yes,


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everything; nowadays we think--nothing. Already thedistance-concept is logically arbitrary; there need be nothings that correspond to it, even approximately.Something similar may be said of the concepts straightline, plane, of three-dimensionality and of the validity of Pythagoras' theorem. Nay, even the continuum-doctrineis in no wise given with the nature of human thought, sothat from the epistemological point of view no greaterauthority attaches to the purely topological relationsthan to the others.

Earlier Physical Concepts. --We have yet to deal withthose modifications in the space-concept, which haveaccompanied the advent of the theory of relativity. Forthis purpose we must consider the space-concept of theearlier physics from a point of view different from thatabove. If we apply the theorem of Pythagoras toinfinitely near points, it reads

ds = dx + dy + dz

where ds denotes the measurable interval betweenthem. For an empirically-given ds the co-ordinate systemis not yet fully determined for every combination of points by this equation. Besides being translated, a co-ordinate system may also be rotated. This signifiesanalytically: the relations of Euclidean geometry arecovariant with respect to linear orthogonaltransformations of the co-ordinates.

Change of direction of the co-ordinate axes while theirorthogonality is preserved.

In applying Euclidean geometry to pre-relativisticmechanics a further indeterminateness enters throughthe choice of the co-ordinate system: the state of motionof the co-ordinate system is arbitrary to a certaindegree, namely, in that substitutions of the co-ordinatesof the form

x' = x - vt

y' = y


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z' = z

also appear possible. On the other hand, earliermechanics did not allow co-ordinate systems to beapplied of which the states of motion were different fromthose expressed in these equations. In this sense wespeak of "inertial systems." In these favoured-inertialsystems we are confronted with a new property of spaceso far as geometrical relations are concerned. Regardedmore accurately, this is not a property of space alonebut of the four-dimensional continuum consisting of timeand space conjointly.

Appearance of Time. --At this point time enters explicitlyinto our discussion for the first time. In their applicationsspace (place) and time always occur together. Everyevent that happens in the world is determined by thespace-co-ordinates x, y, z, and the time-co-ordinate t.

Thus the physical description was four-dimensional rightfrom the beginning. But this four-dimensional continuumseemed to resolve itself into the three-dimensionalcontinuum of space and the one-dimensional continuumof time. This apparent resolution owed its origin to theillusion that the meaning of the concept "simultaneity" isself-evident, and this illusion arises from the fact that wereceive news of near events almost instantaneouslyowing to the agency of light.

This faith in the absolute significance of simultaneitywas destroyed by the law regulating the propagation of light in empty space or, respectively, by the Maxwell-Lorentz electrodynamics. Two infinitely near points canbe connected by means of a light-signal if the relation

ds = c dt - dx - dy - dz = 0

holds for them. It further follows that ds has a valuewhich, for arbitrarily chosen infinitely near space-timepoints, is independent of the particular inertial systemselected. In agreement with this we find that for passingfrom one inertial system to another, linear equations of transformation hold which do not in general leave thetime-values of the events unchanged. It thus becamemanifest that the four-dimensional continuum of space


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cannot be split up into a time-continuum and a space-continuum except in an arbitrary way. This invariantquantity ds may be measured by means of measuring-rods and clocks.

Four-Dimensional Geometry. --On the invariant ds a four-dimensional geometry may be built up which is in alarge measure analogous to Euclidean geometry in threedimensions. In this way physics becomes a sort of staticsin a four-dimensional continuum. Apart from thedifference in the number of dimensions the lattercontinuum is distinguished from that of Euclideangeometry in that ds may be greater or less than zero.Corresponding to this we differentiate between time-likeand space-like line-elements. The boundary betweenthem is marked out by the element of the "light-cone" ds

= 0 which starts out from every point. If we consideronly elements which belong to the same time-value, wehave

- ds = dx + dy + dz

These elements ds may have real counterparts indistances at rest and, as before, Euclidean geometryholds for these elements.

Effect of Relativity, Special and General. --This is themodification which the doctrine of space and time hasundergone through the restricted theory of relativity.

The doctrine of space has been still further modified bythe general theory of relativity, because this theorydenies that the three-dimensional spatial section of thespace-time continuum is Euclidean in character.

Therefore it asserts that Euclidean geometry does nothold for the relative positions of bodies that arecontinuously in contact.

For the empirical law of the equality of inertial andgravitational mass led us to interpret the state of thecontinuum, in so far as it manifests itself with referenceto a non-inertial system, as a gravitational field and totreat non-inertial systems as equivalent to inertialsystems. Referred to such a system, which is connectedwith the inertial system by a non-linear transformation of


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the co-ordinates, the metrical invariant ds assumes thegeneral form:--

ds = g{sub }dx dx

where the g{sub }'s are functions of the co-ordinatesand where the sum is to be taken over the indices for allcombinations 11, 12, . . . 44. The variability of the g{sub

}'s is equivalent to the existence of a gravitationalfield. If the gravitational field is sufficiently general it isnot possible at all to find an inertial system, that is, a co-ordinate system with reference to which ds may beexpressed in the simple form given above:--

ds = c dt - dx - dy - dz

But in this case, too, there is in the infinitesimalneighbourhood of a space-time point a local system of reference for which the last-mentioned simple form fords holds.

This state of the facts leads to a type of geometry whichRiemann's genius created more than half a centurybefore the advent of the general theory of relativity of which Riemann divined the high importance for physics.

Riemann's Geometry. --Riemann's geometry of an n-dimensional space bears the same relation to Euclideangeometry of an n-dimensional space as the generalgeometry of curved surfaces bears to the geometry of the plane. For the infinitesimal neighbourhood of a pointon a curved surface there is a local co-ordinate systemin which the distance ds between two infinitely nearpoints is given by the equation

ds = dx + dy

For any arbitrary (Gaussian) co-ordinate-system,however, an expression of the form

ds = g{sub 11}dx + 2g{sub 12}dx dx + g{sub 22}dx

holds in a finite region of the curved surface. If the


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g{sub }'s are given as functions of x and x thesurface is then fully determined geometrically. For fromthis formula we can calculate for every combination of two infinitely near points on the surface the length ds of the minute rod connecting them; and with the help of this formula all networks that can be constructed on thesurface with these little rods can be calculated. Inparticular, the "curvature" at every point of the surfacecan be calculated; this is the quantity that expresses towhat extent and in what way the laws regulating thepositions of the minute rods in the immediate vicinity of the point under consideration deviate from those of thegeometry of the plane.

This theory of surfaces by Gauss has been extended byRiemann to continua of any arbitrary number of dimensions and has thus paved the way for the generaltheory of relativity. For it was shown above thatcorresponding to two infinitely near space-time pointsthere is a number ds which can be obtained bymeasurement with rigid measuring-rods and clocks (inthe case of time-like elements, indeed, with a clockalone). This quantity occurs in the mathematical theoryin place of the length of the minute rods in three-dimensional geometry. The curves for which ds hasstationary values determine the paths of material pointsand rays of light in the gravitational field, and the"curvature" of space is dependent on the matterdistributed over space.

Just as in Euclidean geometry the space-concept refersto the position-possibilities of rigid bodies, so in thegeneral theory of relativity the space-time-conceptrefers to the behaviour of rigid bodies and clocks. Butthe space-time-continuum differs from the space-continuum in that the laws regulating the behaviour of these objects (clocks and measuring-rods) depend onwhere they happen to be. The continuum (or thequantities that describe it) enters explicitly into the lawsof nature, and conversely these properties of thecontinuum are determined by physical factors. Therelations that connect space and time can no longer bekept distinct from physics proper.


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Nothing certain is known of what the properties of thespace-time-continuum may be as a whole. Through thegeneral theory of relativity, however, the view that thecontinuum is infinite in its time-like extent but finite inits space-like extent has gained in probability.

TIME

The physical time-concept answers to the time-conceptof the extra-scientific mind. Now, the latter has its rootin the time-order of the experiences of the individual,and this order we must accept as something primarilygiven.

I experience the moment "now," or, expressed moreaccurately, the present sense-experience ( Sinnen-Erlebnis ) combined with the recollection of (earlier)sense-experiences. That is why the sense-experiencesseem to form a series, namely the time-series indicatedby "earlier" and "later." The experience-series is thoughtof as a one-dimensional continuum. Experience-seriescan repeat themselves and can then be recognised.

They can also be repeated inexactly, wherein someevents are replaced by others without the character of the repetition becoming lost for us. In this way we formthe time-concept as a one-dimensional frame which canbe filled in by experiences in various ways. The sameseries of experiences answer to the same subjectivetime-intervals.

The transition from this "subjective" time ( Ich-Zeit ) tothe time-concept of pre-scientific thought is connectedwith the formation of the idea that there is a realexternal world independent of the subject. In this sensethe (objective) event is made to correspond with thesubjective experience. In the same sense there isattributed to the "subjective" time of the experience a"time" of the corresponding "objective" event. Incontrast with experiences external events and theirorder in time claim validity for all subjects.

This process of objectification would encounter nodifficulties were the time-order of the experiencescorresponding to a series of external events the same


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for all individuals. In the case of the immediate visualperceptions of our daily lives, this correspondence isexact. That is why the idea that there is an objectivetime-order became established to an extraordinaryextent. In working out the idea of an objective world of external events in greater detail, it was found necessaryto make events and experiences depend on each otherin a more complicated way. This was at first done bymeans of rules and modes of thought instinctivelygained, in which the conception of space plays aparticularly prominent part. This process of refinementleads ultimately to natural science.

The measurement of time is effected by means of clocks. A clock is a thing which automatically passes insuccession through a (practically) equal series of events(period). The number of periods (clock-time) elapsedserves as a measure of time. The meaning of thisdefinition is at once clear if the event occurs in theimmediate vicinity of the clock in space; for all observersthen observe the same clock-time simultaneously withthe event (by means of the eye) independently of theirposition. Until the theory of relativity was propounded itwas assumed that the conception of simultaneity had anabsolute objective meaning also for events separated inspace.

This assumption was demolished by the discovery of thelaw of propagation of light. For if the velocity of light inempty space is to be a quantity that is independent of the choice (or, respectively, of the state of motion) of the inertial system to which it is referred, no absolutemeaning can be assigned to the conception of thesimultaneity of events that occur at points separated bya distance in space. Rather, a special time must beallocated to every inertial system. If no co-ordinatesystem (inertial system) is used as a basis of referencethere is no sense in asserting that events at differentpoints in space occur simultaneously. It is inconsequence of this that space and time are weldedtogether into a uniform four-dimensional continuum.

(A.E.)

EspacioTiempo - A Einstein

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