It has been shown that the extension of the elasticity theory in more than
three dimensions allows a description of space-time as a properly stressed
medium, even recovering the Minkowski metric in the case of uniaxial stress.
The fundamental equation for the metric in the theory is shown to be the
equilibrium equation for the medium. Examples of spherical and cylindrical
symmetries in four dimensions are considered, evidencing convergencies and
divergencies with the classical general relativity theory. Finally the possible
meaning of the dynamics of the four dimensional elastic medium is discussed.Comment: 10 pages; LATEX, uses thmsa.sty now sent to enable PS conversion; to
  appear on Gravitation & Cosmolog

Tartaglia, A.

English

arXiv

TARTAGLIA, Angelo

PORTO@iris (Publications Open Repository TOrino - Politecnico di Torino)

05 August 2020POLITECNICO DI TORINORepository ISTITUZIONALEFourdimensional Elasticity and General Relativity / TARTAGLIA A.. - In: GRAVITATION & COSMOLOGY. - ISSN 0202-2893. - 1(1995), pp. 335-337.OriginalFourdimensional Elasticity and General RelativityPublisher:PublishedDOI:Terms of use:openAccessPublisher copyright(Article begins on next page)This article is made available under terms and conditions as specified in the  corresponding bibliographic description inthe repositoryAvailability:This version is available at: 11583/1406726 since:MAIK Nauka/InterperiodicaGravitation €3 Cosmology, VoI. 1 (1995), No.4, pp. 335-338© 1995 Russian Gravitational SocietyFOUR-DIMENSIONAL ELASTICITY AND GENERAL RELATIVITYA. Tartaglia'Dip. Fisica, Politecnico, Corso Duca degli Abruzzi 24, 1-10129 Torino, Italy.Received 07 February 1995Received in final form 28 July 1995It has been shown that an extension of the elasticity theory to more than three dimensions enables one to describethe space-time as a properly stressed medium, even recovering the Minkowski metric in the case of uniaxial stress. AIundamental equation for the metric in the theory is shown to be the equilibrium equation for the medium. Examplesof spherical and cylindrical symmetries in four dimensions are considered, indicating convergencies and divergencieswith classical general relativity. Finally, a possible meaning of the dynamics of four-dimensional elastic medium isdiscussed.1. IntroductionThe tensor theory of elasticity in three dimensions hassome apparent similarities with classical generaI rel-ativity. The question I have addressed in this paperis. whether or not this formaI analogy may correspondto something more profound than a mere use of sym-metri c tensors in both cases. In fact. many authorshave tried and introduced elasticity into general rela-tivity, casting the general equations into a relativisticform [1-6]. This was usually made for "practical" pur-poses, in order to describe the dynarnic behaviour ofastrophysical bodies in relativistic conditions, the in-teraction of gravitational waves with bar antennas, thepropagation of shock waves in viscoelastic media andthe like.Here the approach is different, because the space-time itself will be looked at as an elastic medium. Somehint in that direction can be found in the literature, forinstance, in [7] where Gerlach and Scott introduce asort of "elasticity of the metric" , though in connectionwith the presence of matter.The guiding idea of this paper is as follows: supposethe space-time is a four dimensional elastic medium.The latter, when unstrained, is perfectly homogeneousand isotropic. The fundamental symmetry around anypoint inside it is GL(4,R). Apply now some stress-es to the medium: the symmetry will be broken andreduced. In particular, if the applied stress is one-dimensional, the consequence is that one particular di-rection is specialized: could this be time? Otherwisestated: is it possible that a uniaxial stress reduces theGL(4, R) symmetry to SO(3, l)?This approach, as we shall see, is indeed viable atl e-mail: tartaglia@itopoli.bitneta first level but leads (I would say 'of course') to finalequilibrium equations which are different from thoseof generaI relativity. First of all, the linear theory ofelasticity in any number of dimensions leads inevitablyto linear equations. The description of space-time thatcomes out as a result is "static", i.e., perfectly deter-ministic. Any dynamics of a four-dimensional mediumneeds a fifth evolution parameter and the evolution it-self would bring about certain modifications both inthe past and the future of a given eventoIn what follows l shall review the theory of elas-ticity and show how it can be brought to describe areasonable space-time.2. Instant review of elasticity theory2.1. Str ainSuppose you have an n-dimensional elastic medium.In the absence of any strain the geometry inside it isEuclidean (or at least, assume it is). The squared dis-tance between two nearby points is(1)where fJ.lv is the metric tensor of the unstrained medi-um.Introduce now an infinitesimal strain. Any pointwill be, in general, displaced by a small vector tU.varying from place to place. As a consequence the newsquared distance between two points will beThe quantities x stilI refer to the unperturbed back-ground and the tU, as well as e , are functions of the1336 A. Tartagliaxa. In an explicit form one has:OEa{3 == Eaf3,I"WI",owa = w~dxl"where commas denote par-tial derivati ves.Now Eq. (2) may be written in the formdl2 = dxadxf3(Eaf3 + Ea!JWfl + EI"{3W;a+ I" v+ 1"+ v I"El"vW,aW,f3 Ea{3,I"W Eal",vW w,{3+ VI"+ ÀI"V)EI"f3,vW w,a El"v,ÀW w,aw,{3 .Usually a part of the content of the brackets in(4) is identified with the strain tensor uI"V, which ismanifestly symmetric.Now (4) becomesdl2 = (fa8 + u(8)dxadxf3. (5)The almost obvious identificationleads todl2 = gl"vdxl"dxv.The symmetric tensor gl"v is now the metri, tensor ofthe strained medium.Ali this, as said, has been written using the unper-turbed Euclidean coordinates. It is more natural tohave recourse to internaI or intrinsic coordinates (thoseattached to the medium); these (let us cali them f,ILwill in generaI be functions of x!". The geornetr ic na-ture of the objects used in the theory is such that it ispossible to recast everything in terms of E,I' by stan-dard coordinate transformations for tensors. In prac-tice we can simply rewrite forrnulas from (5) to (7) asif the xl" were f,I" and nothing changes, consequently,we shall continue to use XIL in the new meaning [8J.By t/w way. in the base situation (unstrained medium)x!' anel f,I" coincide.2.2. StressIn the classical theory of elasticity, a stress ten-sor , whose element (Jaf3 has the meaning of the (Y-component of the force per unit surface acting on asurface e/ement orthogonal to the 8-direction, is in-troduced. Assuming the range of stresses to be zero(propagation only by surface interaction), (JI"V is sym-metrico The next step is to link stresses and strains.This can be done by the so-called Hooke's law, whichis indeed a linearization stating the proportionality be-tween stresses and strains. The theory may be foundin any textbook on elasticity, such as, for instance,[9J. Hooke's law in n dimensions is expressed by theequivalent formulas(2j.l) vÀ(Ja{3 = K - ~ Eaf3( UÀv + 2j.lua{3,ua{3 = C)K - 2~n) Ea{3EvÀ(JvÀ + 2~(Jaf3 (8)(3)(4)(6)(7)where K is the uniform compression modulus and j.l isthe shear modulus. Reasonable restrictions upon thevalues of K and j.l are"u » O. (9)K > O,In generaI, in a linearized theory of elasticity in anydimensions two independent parameters are sufficientfor describing the properties of the rnediurn. The pa-rameters used may be variously combined to produceothers such as the first Lamé coefficient À (the secondis j.l), the Young modulus E and the Poisson coeffi-cient (J.3. Equilibrium conditionsIn our homogeneous stressed medium an equilibriumis attained when the following equation holds:(Ja{3,f3+ fa = O ( lO)Now fa represents the o -component of any force perunit volume; due to the linearity of Hooke's law, indicesare raised and lowereel using f I"V .Combining Eqs (8), (lO) ancl (6). one directly ob-tains:[(K - 2IL/n)((vÀgÀv - n) - 21L] f('<fU}+ (!{ - 2JL/n)fa!3(fI"V gJ.lV ),v + 2/Iga!3,t3 = - t.: (Il)In Cartesian coordinates (with the Eucliclean back-ground geometry), (11) is simplified to(12)Eqs.(ll) or (l2) are n equations for n(11. + 1)/2unknowns, consequently, the problem is underdeter-minecl. Suitable boundary conclitions are neecled.4. Uniaxial stressLet us now suppose that in our homogeneous 11.-climensional meclium a uniform stress is applied alongan arbitrary direction and cali the corresponding axisthe T axis. The stress tensor (as referrecl to a Carte-sian coordinates system) is in our conditions(Joo p,o: :f- 8,O,2:. (13)The index ~umber O corresponds to T, Latin indicesrun from l to n-l; 2: and pare constants; p > Omeans traction and p < O means compression.Looking at Eq. (12), we see that any s" = constis a solution of the equilibriurn equation. To actually2Four-Dimensionol Elasticity and Generai Relativity 337solve the problem, we have to directly deduce gJ.'lI from(6) and Hooke's law (8):.!.[(_1+n-l)p+(n_l)(_l -~).E],n nI< 2/l «« 2flO, a =f 13,l [(/ 1 1 ) (n-l 1)]- - - - p + - + - .E. (14)n «« 2fl «« 2flUooUi;Now applying (6), we see that gllll = "'Ill/' where"'Ill/ is the Minkowski metri c tensor, wheneverP.E[(n - 1)/n](2fl- nI<),-[2fl + nI«n - l)]n .In four dimensions that is:P.E!(Il - 2I<),-~(fl + 6I<).In view of the conditions (9), it comes out that.E< O in any case, which means transverse compres-sion. This is consistent with what we know from three-dimensional elasticity if p > O, i.e., if there is tractionalong the r axis. The parameter p is actually positivewhenIl> 21\..The Minkowski space-tirne looks like a four-di-mensional medium with suitable elastic propertiesstretched along the time axis, Before the stress isapplied, there is no difference among various coordi-nates, so there is no "time"; once there is a stress, oneof the coordinates, measured along any axis within thelight cone about r , becomes no longer interchangeablewith the others: this, from the intrinsic viewpoint, istirne.5. Spatially flat expanding umverseAnother case of interest is that of an open expandinguniverse. The corresponding conformally flat metric,in Cartesian coordinates, may be written asIntroducing the metric (18) into (12), it is easilyverified that a nontrivial solution is found for a( r) iffa = F = constoThe solution 1SFa2 (T) = 2I< _ 3fl r + constoand is consistent with the existence of a uniform vol-ume field orthogonal to any space section of the four-dimensional elastic medium,(15)(16)( 17)( 18)(19)After time rescaling according toa(T)dr=dt,the line element acquires the synchronous form(20)with(21)As can be seen, the time dependence of the spacescale factor is the same as that for a matter-dominatedFriedmann universe [l.O].The solution we have found corresponds to a spher-ically symmetric situation in four dimensions. Thereis one center of symmetry (the big bang) and any ra-diai axis may be used as a tirne axis. Unlike that,a more generaI positive or negative space curvatureRobertson- Walker metric does not comply with thissymmetry and is not a solution to Eq. (11).6. Rotation symmetry about an aXISThis is a typical situation whirh in generaI relativityleads, in the stati c case, to thc Schwarzschilel solurion.A generaI form for a metric with this syrnmetry is(J(~ T)g= OO(2:!)The cylindrical coordinates T, T, fJ, 'P bave been used.The correspondiug expression for the e iso1OO(23)Inserting (22) anel (2:3) into (11) leads to a pair of in-dependent equations:.. .(I< - 2Il)(f - h) + 2flf(I< - 2fl)(J' - h') - 2flf'-fo.-fr. (24)Dots stand for partial elerivatives with respect to r andprimes for partial derivatives with respect to r.The static case (r-independence of f and h) re-quires fo = O, whereas nontrivial solutions exist onlyif t- =f O. To actually solve the problem, it is neces-sary to impose the distribution of strains or stresseson a suitable surface, remembering also that f and hshould be positive.One can arrive at the same results starting fromthe solution for the uniaxial stress case and allowing pand .E to depend on T and r.3338 A. Tartaglia7. DiscussionIt has been shown that solutions of the equilibriumconditions inside a four-dimensional elastic mediumstressed in any way may provide reasonable forms forspace-time metric under various symmetry conditions.There are, however , some problems; one ofthem is thatof the signature.Treating the case of a uniaxial stress, we saw thatit is possible to recover the Minkowski metrico In fourdimensions (14) and (16) lead to the strain tensor com-ponentsUoo = 0, Uii = -2.However , we know that the strain tensor is definedstarting from a strain vector fleld according to (4). Inthe case of uniaxial symmetry the explicit form of thestrain tensor in the background Euclidean coordinates1SUIJ.V= wIJ.,V + WV,IJ.+ w~ w""v·Substituting (25) into (26) and solving for the io" , oneobtains:'u'o = const - 27, wr = (-l =Fi)r.Thus, while the strain tensor is real, the strain vec-tor field is complex: this is the price to be paid for theMinkowski signature.Another important point to remind is that the theo-ry, due to Hooke's law, is linear. This implies that onlyweak field regions may be described in this way. It ispossible to obtain a better approximation for strongerfields considering nonlinear elasticity. The startingpoint is the development of the Helmholz free ener-gy F of the medium in powers of the strains, whereHooke's law comes from. The next approximation afterthe linear one isÀ 2 1/ 3F = F', + - (ua) + jJU u/lO! + -( ua)o 2 a a IJ. ;3 . '"+avlJ.+ {31J."'+~UaUIJ.Uv PUaU{3UIJ. ...Three new parameters (1/, it , p) have been intro-duced to characterize the behaviour of the medium.lt is now no longer allowed to raise and lower indicesusing simply the f 's; instead, gl'v must be used afterdeveloping them to the first order in ur . Everythingis rnuch more complicated but it may be managed.Finally we may remark that our treatment of anequilibrium condition corresponds to a perfectly stat-ic situation, i.e., to an entirely deterrninistic universewhere the histories coincide with the flux lines of thestrain vector field. However, any elastic medium hasnot only statics but also dynamics: it may vibrate andhas characteristic internaI frequencies. If we consider afour-dimensional elastic medium, vibrations evidentlyhave a meaning only with respect to some appropri-ate evolution parameter, let us cali it T: something(25)(26)(27)(28)like the good old Newtonian time. Four-dimensionalobservers have of course no means to measure T:their clocks actually measure what we called 7 (or t),though now 7, as welI as the other space coordinates,parametricalIy depends on T. An influence of the vi-brations in four plus one dimensions may, however, beseen from inside the four-dimensional world.Suppose, for instance, that the medium contains acouple of points held fixed, whereas the rest undergoeselastic vibrations. Any strain flux line (or history) go-ing from one point to the other is continuously modifiedby the vibrations in T. An internaI observer, unableto perceive T, will notice that there are many nearbyhistories, and what in four plus one elimensions is aT evolution for him may welI be transformed into dif-ferent probabilities to be att.ached to various histories.If the four-dimensional observer wants to forecast thefuture, he will be led to an average over histories. Aremarkable feature is the fact that for a given vibratingpoint both the future and the past (in 7 or t) vary inT.I think that this viewpoint may provicle a new ap-proach to quantum mechanics anel in aoy case is worthfurther investigation.References[1] J.L. Synge, Math. Z. 72 (1959), 82,[2] C.B. Rayner, Proc. Roy. Soc. Lorulon, 272A (196.3),44.[3] A.D. Sakhar ov, Dokl. Akad. Nouk S'S'S'H. 177 (1967).70.[4] A. Papapetrou, AnTl, lnst. H, Poincaré lA (1972), 63.[5] G. Maugin, Comm, Math. Phy.s. 53 (1977), 23.3,[6] J. Katz and D. Lynden-Bell, elas.s. Quantum Grau. 8(1991),22.3l.[7] V.H. C~erlach and J.F. Scot.t, Plujs, Re» .. D34 (1986),3638.[8] W. Hern andes, Phy.s. Rev., DI (1970), 101.3.[9] L. Landau and E. Lifshitz, Théorie de I'élasticité,Éditions Mir , Moscow, 1967.[tu] W. Misner. K.S. Thorne and J.A. Wheeler. "Gravita-tion", Freem an and Co. San Francisco, 1973, p. 735.4

Fourdimensional Elasticity and General Relativity

Tartaglia, Angelo

PORTO Publications Open Repository TOrino

Politecnico di Torino
Porto Institutional Repository
[Article] Fourdimensional Elasticity and General Relativity
Original Citation:
Tartaglia A. (1995). Fourdimensional Elasticity and General Relativity. In: GRAVITATION &
COSMOLOGY, vol. 1, pp. 335-337. - ISSN 0202-2893
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Gravitation €3 Cosmology, VoI. 1 (1995), No.4, pp. 335-338
© 1995 Russian Gravitational Society
FOUR-DIMENSIONAL ELASTICITY AND GENERAL RELATIVITY
A. Tartaglia'
Dip. Fisica, Politecnico, Corso Duca degli Abruzzi 24, 1-10129 Torino, Italy.
Received 07 February 1995
Received in final form 28 July 1995
It has been shown that an extension of the elasticity theory to more than three dimensions enables one to describe
the space-time as a properly stressed medium, even recovering the Minkowski metric in the case of uniaxial stress. A
Iundamental equation for the metric in the theory is shown to be the equilibrium equation for the medium. Examples
of spherical and cylindrical symmetries in four dimensions are considered, indicating convergencies and divergencies
with classical general relativity. Finally, a possible meaning of the dynamics of four-dimensional elastic medium is
discussed.
1. Introduction
The tensor theory of elasticity in three dimensions has
some apparent similarities with classical generaI rel-
ativity. The question I have addressed in this paper
is. whether or not this formaI analogy may correspond
to something more profound than a mere use of sym-
metri c tensors in both cases. In fact. many authors
have tried and introduced elasticity into general rela-
tivity, casting the general equations into a relativistic
form [1-6]. This was usually made for "practical" pur-
poses, in order to describe the dynarnic behaviour of
astrophysical bodies in relativistic conditions, the in-
teraction of gravitational waves with bar antennas, the
propagation of shock waves in viscoelastic media and
the like.
Here the approach is different, because the space-
time itself will be looked at as an elastic medium. Some
hint in that direction can be found in the literature, for
instance, in [7] where Gerlach and Scott introduce a
sort of "elasticity of the metric" , though in connection
with the presence of matter.
The guiding idea of this paper is as follows: suppose
the space-time is a four dimensional elastic medium.
The latter, when unstrained, is perfectly homogeneous
and isotropic. The fundamental symmetry around any
point inside it is GL(4,R). Apply now some stress-
es to the medium: the symmetry will be broken and
reduced. In particular, if the applied stress is one-
dimensional, the consequence is that one particular di-
rection is specialized: could this be time? Otherwise
stated: is it possible that a uniaxial stress reduces the
GL(4, R) symmetry to SO(3, l)?
This approach, as we shall see, is indeed viable at
l e-mail: tartaglia@itopoli.bitnet
a first level but leads (I would say 'of course') to final
equilibrium equations which are different from those
of generaI relativity. First of all, the linear theory of
elasticity in any number of dimensions leads inevitably
to linear equations. The description of space-time that
comes out as a result is "static", i.e., perfectly deter-
ministic. Any dynamics of a four-dimensional medium
needs a fifth evolution parameter and the evolution it-
self would bring about certain modifications both in
the past and the future of a given evento
In what follows l shall review the theory of elas-
ticity and show how it can be brought to describe a
reasonable space-time.
2. Instant review of elasticity theory
2.1. Str ain
Suppose you have an n-dimensional elastic medium.
In the absence of any strain the geometry inside it is
Euclidean (or at least, assume it is). The squared dis-
tance between two nearby points is
(1)
where fJ.lv is the metric tensor of the unstrained medi-
um.
Introduce now an infinitesimal strain. Any point
will be, in general, displaced by a small vector tU.
varying from place to place. As a consequence the new
squared distance between two points will be
The quantities x stilI refer to the unperturbed back-
ground and the tU, as well as e , are functions of the
1
336 A. Tartaglia
xa. In an explicit form one has:
OEa{3 == Eaf3,I"WI",
owa = w~dxl"
where commas denote par-tial derivati ves.
Now Eq. (2) may be written in the form
dl
2 = dxadxf3(Eaf3 + Ea!JWfl + EI"{3W;a
+ I" v+ 1"+ v I"El"vW,aW,f3 Ea{3,I"W Eal",vW w,{3
+ VI"+ ÀI"V)EI"f3,vW w,a El"v,ÀW w,aw,{3 .
Usually a part of the content of the brackets in
(4) is identified with the strain tensor uI"V, which is
manifestly symmetric.
Now (4) becomes
dl2 = (fa8 + u(8)dxadxf3. (5)
The almost obvious identification
leads to
dl2 = gl"vdxl"dxv.
The symmetric tensor gl"v is now the metri, tensor of
the strained medium.
Ali this, as said, has been written using the unper-
turbed Euclidean coordinates. It is more natural to
have recourse to internaI or intrinsic coordinates (those
attached to the medium); these (let us cali them f,IL
will in generaI be functions of x!". The geornetr ic na-
ture of the objects used in the theory is such that it is
possible to recast everything in terms of E,I' by stan-
dard coordinate transformations for tensors. In prac-
tice we can simply rewrite forrnulas from (5) to (7) as
if the xl" were f,I" and nothing changes, consequently,
we shall continue to use XIL in the new meaning [8J.
By t/w way. in the base situation (unstrained medium)
x!' anel f,I" coincide.
2.2. Stress
In the classical theory of elasticity, a stress ten-
sor , whose element (Jaf3 has the meaning of the (Y-
component of the force per unit surface acting on a
surface e/ement orthogonal to the 8-direction, is in-
troduced. Assuming the range of stresses to be zero
(propagation only by surface interaction), (JI"V is sym-
metrico The next step is to link stresses and strains.
This can be done by the so-called Hooke's law, which
is indeed a linearization stating the proportionality be-
tween stresses and strains. The theory may be found
in any textbook on elasticity, such as, for instance,
[9J. Hooke's law in n dimensions is expressed by the
equivalent formulas
(
2j.l) vÀ
(Ja{3 = K - ~ Eaf3( UÀv + 2j.lua{3,
ua{3 = C)K - 2~n) Ea{3EvÀ(JvÀ + 2~(Jaf3 (8)
(3)
(4)
(6)
(7)
where K is the uniform compression modulus and j.l is
the shear modulus. Reasonable restrictions upon the
values of K and j.l are
"u » O. (9)K > O,
In generaI, in a linearized theory of elasticity in any
dimensions two independent parameters are sufficient
for describing the properties of the rnediurn. The pa-
rameters used may be variously combined to produce
others such as the first Lamé coefficient À (the second
is j.l), the Young modulus E and the Poisson coeffi-
cient (J.
3. Equilibrium conditions
In our homogeneous stressed medium an equilibrium
is attained when the following equation holds:
(Ja{3,f3+ fa = O ( lO)
Now fa represents the o -component of any force per
unit volume; due to the linearity of Hooke's law, indices
are raised and lowereel using f I"V .
Combining Eqs (8), (lO) ancl (6). one directly ob-
tains:
[(K - 2IL/n)((vÀgÀv - n) - 21L] f('<fU}
+ (!{ - 2JL/n)fa!3(fI"V gJ.lV ),v + 2/Iga!3,t3 = - t.: (Il)
In Cartesian coordinates (with the Eucliclean back-
ground geometry), (11) is simplified to
(12)
Eqs.(ll) or (l2) are n equations for n(11. + 1)/2
unknowns, consequently, the problem is underdeter-
minecl. Suitable boundary conclitions are neecled.
4. Uniaxial stress
Let us now suppose that in our homogeneous 11.-
climensional meclium a uniform stress is applied along
an arbitrary direction and cali the corresponding axis
the T axis. The stress tensor (as referrecl to a Carte-
sian coordinates system) is in our conditions
(Joo p,
o: :f- 8,O,
2:. (13)
The index ~umber O corresponds to T, Latin indices
run from l to n-l; 2: and pare constants; p > O
means traction and p < O means compression.
Looking at Eq. (12), we see that any s" = const
is a solution of the equilibriurn equation. To actually
2
Four-Dimensionol Elasticity and Generai Relativity 337
solve the problem, we have to directly deduce gJ.'lI from
(6) and Hooke's law (8):
.!.[(_1+n-l)p+(n_l)(_l -~).E],
n nI< 2/l «« 2fl
O, a =f 13,
l [(/ 1 1 ) (n-l 1)]- - - - p + - + - .E. (14)
n «« 2fl «« 2fl
Uoo
Ui;
Now applying (6), we see that gllll = "'Ill/' where
"'Ill/ is the Minkowski metri c tensor, whenever
P
.E
[(n - 1)/n](2fl- nI<),
-[2fl + nI«n - l)]n .
In four dimensions that is:
P
.E
!(Il - 2I<),
-~(fl + 6I<).
In view of the conditions (9), it comes out that
.E< O in any case, which means transverse compres-
sion. This is consistent with what we know from three-
dimensional elasticity if p > O, i.e., if there is traction
along the r axis. The parameter p is actually positive
when
Il> 21\..
The Minkowski space-tirne looks like a four-di-
mensional medium with suitable elastic properties
stretched along the time axis, Before the stress is
applied, there is no difference among various coordi-
nates, so there is no "time"; once there is a stress, one
of the coordinates, measured along any axis within the
light cone about r , becomes no longer interchangeable
with the others: this, from the intrinsic viewpoint, is
tirne.
5. Spatially flat expanding umverse
Another case of interest is that of an open expanding
universe. The corresponding conformally flat metric,
in Cartesian coordinates, may be written as
Introducing the metric (18) into (12), it is easily
verified that a nontrivial solution is found for a( r) if
fa = F = consto
The solution 1S
F
a
2 (T) = 2I< _ 3fl r + consto
and is consistent with the existence of a uniform vol-
ume field orthogonal to any space section of the four-
dimensional elastic medium,
(15)
(16)
( 17)
( 18)
(19)
After time rescaling according to
a(T)dr=dt,
the line element acquires the synchronous form
(20)
with
(21)
As can be seen, the time dependence of the space
scale factor is the same as that for a matter-dominated
Friedmann universe [l.O].
The solution we have found corresponds to a spher-
ically symmetric situation in four dimensions. There
is one center of symmetry (the big bang) and any ra-
diai axis may be used as a tirne axis. Unlike that,
a more generaI positive or negative space curvature
Robertson- Walker metric does not comply with this
symmetry and is not a solution to Eq. (11).
6. Rotation symmetry about an aXIS
This is a typical situation whirh in generaI relativity
leads, in the stati c case, to thc Schwarzschilel solurion.
A generaI form for a metric with this syrnmetry is
(
J(~ T)
g= O
O
(2:!)
The cylindrical coordinates T, T, fJ, 'P bave been used.
The correspondiug expression for the e is
o
1
O
O
(23)
Inserting (22) anel (2:3) into (11) leads to a pair of in-
dependent equations:
.. .
(I< - 2Il)(f - h) + 2flf
(I< - 2fl)(J' - h') - 2flf'
-fo.
-fr. (24)
Dots stand for partial elerivatives with respect to r and
primes for partial derivatives with respect to r.
The static case (r-independence of f and h) re-
quires fo = O, whereas nontrivial solutions exist only
if t- =f O. To actually solve the problem, it is neces-
sary to impose the distribution of strains or stresses
on a suitable surface, remembering also that f and h
should be positive.
One can arrive at the same results starting from
the solution for the uniaxial stress case and allowing p
and .E to depend on T and r.
3
338 A. Tartaglia
7. Discussion
It has been shown that solutions of the equilibrium
conditions inside a four-dimensional elastic medium
stressed in any way may provide reasonable forms for
space-time metric under various symmetry conditions.
There are, however , some problems; one ofthem is that
of the signature.
Treating the case of a uniaxial stress, we saw that
it is possible to recover the Minkowski metrico In four
dimensions (14) and (16) lead to the strain tensor com-
ponents
Uoo = 0, Uii = -2.
However , we know that the strain tensor is defined
starting from a strain vector fleld according to (4). In
the case of uniaxial symmetry the explicit form of the
strain tensor in the background Euclidean coordinates
1S
UIJ.V= wIJ.,V + WV,IJ.+ w~ w""v·
Substituting (25) into (26) and solving for the io" , one
obtains:
'u'o = const - 27, wr = (-l =Fi)r.
Thus, while the strain tensor is real, the strain vec-
tor field is complex: this is the price to be paid for the
Minkowski signature.
Another important point to remind is that the theo-
ry, due to Hooke's law, is linear. This implies that only
weak field regions may be described in this way. It is
possible to obtain a better approximation for stronger
fields considering nonlinear elasticity. The starting
point is the development of the Helmholz free ener-
gy F of the medium in powers of the strains, where
Hooke's law comes from. The next approximation after
the linear one is
À 2 1/ 3F = F', + - (ua) + jJU u/lO! + -( ua)o 2 a a IJ. ;3 . '"
+
avlJ.+ {31J."'+
~UaUIJ.Uv PUaU{3UIJ. ...
Three new parameters (1/, it , p) have been intro-
duced to characterize the behaviour of the medium.
lt is now no longer allowed to raise and lower indices
using simply the f 's; instead, gl'v must be used after
developing them to the first order in ur . Everything
is rnuch more complicated but it may be managed.
Finally we may remark that our treatment of an
equilibrium condition corresponds to a perfectly stat-
ic situation, i.e., to an entirely deterrninistic universe
where the histories coincide with the flux lines of the
strain vector field. However, any elastic medium has
not only statics but also dynamics: it may vibrate and
has characteristic internaI frequencies. If we consider a
four-dimensional elastic medium, vibrations evidently
have a meaning only with respect to some appropri-
ate evolution parameter, let us cali it T: something
(25)
(26)
(27)
(28)
like the good old Newtonian time. Four-dimensional
observers have of course no means to measure T:
their clocks actually measure what we called 7 (or t),
though now 7, as welI as the other space coordinates,
parametricalIy depends on T. An influence of the vi-
brations in four plus one dimensions may, however, be
seen from inside the four-dimensional world.
Suppose, for instance, that the medium contains a
couple of points held fixed, whereas the rest undergoes
elastic vibrations. Any strain flux line (or history) go-
ing from one point to the other is continuously modified
by the vibrations in T. An internaI observer, unable
to perceive T, will notice that there are many nearby
histories, and what in four plus one elimensions is a
T evolution for him may welI be transformed into dif-
ferent probabilities to be att.ached to various histories.
If the four-dimensional observer wants to forecast the
future, he will be led to an average over histories. A
remarkable feature is the fact that for a given vibrating
point both the future and the past (in 7 or t) vary in
T.
I think that this viewpoint may provicle a new ap-
proach to quantum mechanics anel in aoy case is worth
further investigation.
References
[1] J.L. Synge, Math. Z. 72 (1959), 82,
[2] C.B. Rayner, Proc. Roy. Soc. Lorulon, 272A (196.3),
44.
[3] A.D. Sakhar ov, Dokl. Akad. Nouk S'S'S'H. 177 (1967).
70.
[4] A. Papapetrou, AnTl, lnst. H, Poincaré lA (1972), 63.
[5] G. Maugin, Comm, Math. Phy.s. 53 (1977), 23.3,
[6] J. Katz and D. Lynden-Bell, elas.s. Quantum Grau. 8
(1991),22.3l.
[7] V.H. C~erlach and J.F. Scot.t, Plujs, Re» .. D34 (1986),
3638.
[8] W. Hern andes, Phy.s. Rev., DI (1970), 101.3.
[9] L. Landau and E. Lifshitz, Théorie de I'élasticité,
Éditions Mir , Moscow, 1967.
[tu] W. Misner. K.S. Thorne and J.A. Wheeler. "Gravita-
tion", Freem an and Co. San Francisco, 1973, p. 735.
4


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