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Acta Mech (2008)
DOI 10.1007/s00707-013-0973-4
I-Shih Liu ·Rubens Sampaio
Remarks on material frame-indifference controversy
Received: 10 February 2013 / Revised: 5 July 2013
© Springer-Verlag Wien 2013
Abstract Regarding a frame of reference as an observer, the notion of frame-indifference (also referred to as
objectivity) concerns the transformation properties of physical quantities under change of observer. For kine-
matic quantities, they can usually be derived from the deformation/motion, while for non-kinematic quantities,
such as force and stress, A frame-indifference property must be postulated. frame-indifference postulate for
the stress, sometimes unsuitably called the principle of frame-indifference, is a universal assumption, which
has nothing to do with material properties. This has caused some confusions in the interpretation of “material”
frame-indifference in the literature. Material frame-indifference deals with the constitutive functions, which
characterize intrinsic properties of the material under different observers. We shall carefully render these con-
cepts mathematically and deduce the well-known condition of material objectivity as a consequence of the
frame-indifference postulate and the principle of material frame-indifference. We shall also emphasize and
remark on some persistent controversy and some misleading statements found in recent literature.
1 Introduction
The foundation of Modern Continuum Mechanics is based on the fundamental ideas, set forth in the biblical
treatise, The Nonlinear Field Theories of Mechanics, by Truesdell and Noll [17], among which one of the most
important and controversial ones is the principle of material frame-indifference (MFI). Due to the original
somewhat loose statements, attempts for better interpretation of MFI appeared again and again in the literature
and some controversy persisted to these days [1,2,4,11,13,16,18].
As we understand, the essential meaning of MFI is the simple idea that material properties are independent
of observers. In order to explain this, one has to know what a frame of reference (regarded as an observer) is, so
as to define what frame-indifference means. However, frame-indifference concerns only transformation prop-
erties of physical quantities under change of frame, and hence has nothing to do with the material properties.
On the other hand, material frame-indifference, which deals with constitutive functions characterizing intrinsic
properties of the material under different observers, is postulated as a guiding principle for the formulation of
constitutive models. Therefore, frame-indifference and material frame-indifference are two different concepts
not to be confused. We shall make these different concepts mathematically clear in this paper.
A similar version of this paper was included in a conference proceedings [8]. Nevertheless, the present one
has been more carefully refined, and it is one of our objectives to make additional remarks and comments on
some persistent controversy and misleading statements found in recent literature.
I-Shih Liu (B
)
Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil
E-mail: liu@im.ufrj.br
R. Sampaio
Pontifícia Universidade Católica do Rio de Janeiro, Rio de Janeiro, Brazil
2I.-S. Liu, R. Sampaio
Conventional notations now widely used in continuum mechanics textbooks will be followed. For consid-
eration of allowing different observers possibly belong to different Euclidean spaces as sometimes suggested
[11], our formulations take this into consideration, for which we introduce some notions of isometries in
Euclidean spaces with changes in orientation as well as scaling in the “Appendix”, in which examples are
given to justify the rational for such considerations. Please also refer to the “Appendix” for relevant definitions
and notations concerning isometric transformations.
2 Frame of reference
The event world Wis a four-dimensional space–time in which physical events occur at some places and cer-
tain instants. Let Tbe the collection of instants and Wsbe the placement space of simultaneous events at the
instant s, then the neo-classical space–time (Noll [12]) can be expressed as the disjoint union of placement
spaces of simultaneous events at each instant,
W=
s∈T
Ws.
A point ps∈Wis called an event, which occurs at the instant sand the place p∈Ws. At different instants s
and ¯s, the spaces Wsand W¯sare two disjoint spaces. Thus, it is impossible to determine the distance between
two non-simultaneous events at psand p¯sif s=¯s, and hence, Wis not a product space of space and time.
However, it can be set into correspondence with a product space through a frame of reference on W.
Definition (Frame of reference): A frame of reference is a one-to-one mapping
φ:W→E×R,
taking ps→ (x,t),whereRis the space of real numbers and Eis a three-dimensional Euclidean space. We
shall denote the map taking p→ xas the map φs:Ws→E.
In general, the Euclidean spaces of different frames of reference may not be the same. Therefore, for
definiteness, we shall denote the Euclidean space of the frame φby Eφ, and its translation space by Vφ.We
assume that Vφis an inner product space.
Of course, there are infinitely many frames of reference. Each one of them may be regarded as an observer,
since it can be depicted as a person taking a snapshot so that the image of φsis a picture (three dimensional
at least conceptually) of the placements of the events at some instant s, from which the distance between two
simultaneous events can be measured. A sequence of events can also be recorded as video clips depicting the
change of events in time by an observer.
Now, suppose that two observers are recording the same events with video cameras. In order to compare
their video clips regarding the locations and time, they must have a mutual agreement that the clock of their
cameras must be synchronized so that simultaneous events can be recognized, and since during the recording
two observers may move independently while taking pictures with their cameras from different angles, there
will be a relative motion, a scaling and a relative orientation between them. We shall make such a consensus
among observers explicit mathematically.
Let φand φ∗be two frames of reference. They are related by the composite map ∗:=φ∗◦φ−1,∗:
Eφ×R→Eφ∗×R,taking (x,t)→ (x∗,t∗),where(x,t)and (x∗,t∗)are the position and time of the same
event observed by φand φ∗simultaneously. In general, Eφand Eφ∗are different Euclidean spaces. Physically,
not any change of frame would be of our concern as long as we are interested in establishing a consensus
among observers, which should require preservation of distance between simultaneous events and time interval
as well as the sense of time (Fig. 1).
Definition (Euclidean change of frame): A change of frame (observer) from φto φ∗taking (x,t)→ (x∗,t∗)
is an isometry of space and time given by
x∗=Q(t)(x−x0)+c∗(t), t∗=t+a,(1)
for some constant time difference a∈R, some relative translation c∗:R→Eφ∗with respect to the reference
point x0∈Eφand some isometric transformation Q:R→O(Vφ,Vφ∗).
Material frame-indifference controversy 3
Fig. 1 A change of frame
We have denoted O(Vφ,Vφ∗)={Q∈L(Vφ,Vφ∗):QuVφ∗=uVφ,∀u∈Vφ}. Euclidean changes
of frame will often be called changes of frame for simplicity, since they are the only changes of frame among
consenting observers of our concern for the purpose of discussing frame-indifference in continuum mechanics.
All consenting observers form an equivalent class, denoted by E, among the set of all observers, i.e., for any
φ,φ∗∈E, there exists a Euclidean change of frame from φ→φ∗. From now on, only classes of consenting
observers will be considered. Therefore, any observer would mean any observer in some E, and a change of
frame would mean a Euclidean change of frame.
3 Motion and deformation
In the space–time, a physical event is represented by its placement at a certain instant so that it can be observed
in a frame of reference. Let a body Bbe a set of material points.
Definition (Configuration): Let ξ:B→Wtbe a placement of the body Bat the instant t,andletφbe a
frame of reference, then the composite map ξφt:= φt◦ξ,
ξφt:B→Eφ
is called a configuration of the body Bat the instant tin the frame φ.
A configuration thus identifies the body with a region in the Euclidean space of the observer. In this sense,
the motion of a body can be viewed as a continuous sequence of events such that at any instant tthe placement
of the body Bin Wtis a one-to-one mapping
χt:B→Wt,
and the composite mapping χφt:= φt◦χt,
χφt:B→Eφ,x=χφt(p)=φt(χt(p)), p∈B,
is the configuration of the body in the motion χt.LetBχt:= χφt(B)⊂Eφ(see the right part of Fig. 2). The
motion can then be regarded as a sequence of configurations of Bin time, χφ={
χφt,t∈R|χφt:B→Eφ}.
We can also express a motion as
χφ:B×R→Eφ,x=χφ(p,t)=χφt(p), p∈B.
Note that in our discussions we have been using t∈Ras the time in the frame φcorresponding to the instant
s∈Twith s=tfor simplicity without loss of generality.
Fig. 2 Motion χφt, reference configuration κφand deformation χκφ(·,t)
4I.-S. Liu, R. Sampaio
Reference configuration
We regard a body Bas a set of material points. Although it is possible to endow the body as a manifold with a
differentiable structure and topology for doing mathematics on the body, to avoid such mathematical subtleties,
usually a particular configuration is chosen as reference (see the left part of Fig. 2),
κφ:B→Eφ,X=κφ(p), Bκ:= κφ(B)⊂Eφ,
so that the motion at an instant tis a one-to-one mapping
χκφ(·,t):Bκ→Bχt,x=χκφ(X,t)=χφt(κ−1
φ(X)), X∈Bκ,
from a region into another region in the same Euclidean space Eφfor which topology and differentiability
are well defined. This mapping is called a deformation from κto χtin the frame φ, and a motion is then a
sequence of deformations in time.
For the reference configuration κφ, there is some instant, say t0, at which the reference placement of the
body is chosen, κ:B→Wt0(see Fig. 2). On the other hand, the choice of a reference configuration is
arbitrary, and it is not necessary that the body should actually occupy the reference place in its motion under
consideration. Nevertheless, in most practical problems, t0is usually taken as the initial time of the motion.
4 Frame-indifference
The change of frame (1) gives rise to a linear mapping on the translation space, in the following way: Let
u(φ) =x2−x1∈Vφbe the difference vector of x1,x2∈Eφin the frame φ,andu(φ∗)=x∗
2−x∗
1∈Vφ∗
be the corresponding difference vector in the frame φ∗, then from (1), it follows immediately that
u(φ∗)=Q(t)u(φ),
where Q(t)∈O(Vφ,Vφ∗)is the isometric transformation associated with the change of frame φ→φ∗.
Any vector quantity in Vφ, which has this transformation property, is said to be objective with respect to
Euclidean transformations. This concept of objectivity can be generalized to any tensor spaces of Vφ.
Let
s:E→R,u:E→VE,T:E→VE⊗VE,
where Eis the Euclidean class of frames of reference and VE={Vφ:φ∈E}. They are scalar, vector and
(second-order) tensor observable quantities, respectively. We call f(φ) the value of the quantity fobserved
in the frame φ.
Definition (Frame-indifference): Relative to a change of frame from φto φ∗, the observables s,uand T
are called frame-indifferent (or objective) scalar, vector and tensor quantities, respectively, if they satisfy the
following transformation properties:
s(φ∗)=s(φ),
u(φ∗)=Q(t)u(φ),
T(φ∗)=Q(t)T(φ ) Q(t),
where Q(t)∈O(Vφ,Vφ∗)is the isometric transformation of the change of frame from φto φ∗.
More precisely, they are also said to be frame-indifferent with respect to Euclidean transformations or
simply Euclidean objective. For simplicity, we often write f=f(φ) and f∗=f(φ∗).
Material frame-indifference controversy 5
Fig. 3 Reference configurations κφand κφ∗in the change of frame from φto φ∗
Transformation properties of motion
Let χφbe a motion of the body in the frame φ,andχφ∗be the corresponding motion in φ∗,
x=χφ(p,t), x∗=χφ∗(p,t∗), p∈B.
Then, from (1), we have
χφ∗(p,t∗)=Q(t)χφ(p,t)−xo+c∗(t), p∈B,
from which one can easily show that the velocity and the acceleration are not objective quantities,
˙
x∗=Q˙
x+˙
Q(x−xo)+˙
c∗,
¨
x∗=Q¨
x+2˙
Q˙
x+¨
Q(x−x0)+¨
c∗.(2)
A change of frame (1) with constant Qand c∗(t)=c0+c1t, for constant c0and c1(so that ˙
Q=0and¨
c∗=0),
is called a Galilean transformation. Therefore, from (2), we conclude that the acceleration is not Euclidean
objective but it is objective with respect to Galilean transformation. Moreover, it also shows that the velocity
is neither a Euclidean nor a Galilean objective vector quantity.
Transformation properties of deformation gradient
Let κ:B→Wt0be a reference placement of the body at some instant t0(see Fig. 3), then
κφ=φt0◦κand κφ∗=φ∗
t0◦κ(3)
are the corresponding reference configurations of Bin the frames φand φ∗at the same instant, and
X=κφ(p), X∗=κφ∗(p), p∈B.
Let us denote by γ=κφ∗◦κ−1
φthe change of reference configuration from κφto κφ∗in the change of frame,
then it follows from (3)thatγ=φ∗
t0◦φ−1
t0, and by (1), we have
X∗=γ(X)=Q(t0)(X−xo)+c∗(t0). (4)
On the other hand, the motion in referential description relative to the change of frame is given by x=χκ(X,t)
and x∗=χκ∗(X∗,t∗). Hence from (1), we have
χκ∗(X∗,t∗)=Q(t)(χκ(X,t)−xo)+c∗(t).
Therefore, we obtain for the deformation gradients F=∇
XX
Xχκin the frame φand F∗=∇
XX
X∗χκ∗in the frame
φ∗, by the chain rule and the use of (4),
F∗(X∗,t∗)=Q(t)F(X,t)Q(t0),or simply F∗=QFQ
0,(5)
where Q0=Q(t0)is the isometric transformation due to the change of frame at the instant t0when the
reference configuration is chosen.
6I.-S. Liu, R. Sampaio
Remark 1 The transformation property (5) stands in contrast to F∗=QF, the widely used formula that is
obtained “provided that the reference configuration is unaffected by the change of frame” as usually implic-
itly assumed (see [17] p. 308), so that Q0reduces to the identity transformation. On the other hand, the
transformation property F∗=QFQ
0has also been derived elsewhere in the literature ([10,15] Sec. 2.2.8).
Remark 2 Keep in mind that a configuration is a placement of a body relative to a frame of reference and any
reference configuration is not an exception. Therefore, to say that the “reference configuration is unaffected
by the change of frame” is at best an assumption ([2] Sect. 20.1). Oftentimes, in order to justify this assump-
tion, it is remarked that since the transformation Q(t)in the change of frame can be chosen arbitrarily, one
may presume that at some instant t0, it is an identity transformation. This argument may seem quite enticing,
however, it cannot be valid when Qis simply a constant transformation as in the case of Galilean change of
frame.
On the other hand, two consenting observers cannot independently choose their reference configurations,
because they must choose the configuration of the body at the same instant of their respective frames of
reference (see discussions in [5,11]).
5 Galilean invariance of balance laws
In classical mechanics, Newton’s first law, often known as the law of inertia, is essentially a definition of
inertial frame.
Definition (Inertial frame): A frame of reference is called an inertial frame if, relative to it, the velocity of a
body remains constant unless the body is acted upon by an external force.
We present the first law in this manner in order to emphasize that the existence of inertial frames is essential
for the formulation of Newton’s second law, which asserts that relative to an inertial frame, the equation of
motion takes the simple form:
m¨
x=f.(6)
Now, we shall assume that there is an inertial frame φ0∈E, for which the equation of motion of a particle is
given by (6), and we are interested in how the equation is transformed under a change of frame.
Unlike the acceleration, transformation properties of non-kinematic quantities cannot be deduced theoret-
ically. Instead, for the mass mand the force f, it is conventionally postulated that they are Euclidean objective
scalar and vector quantities, respectively, so that for any change from φ0to φ∗∈Egiven by (1), we have
m∗=m,f∗=Qf,
which together with (2), by multiplying (6) with Q, we obtain the equation of motion in the (non-inertial)
frame φ∗,
m∗¨
x∗=f∗+m∗i∗,(7)
where i∗is called the inertial force given by
i∗=¨
c∗+2(˙
x∗−˙
c∗)+(˙
−2)(x∗−c∗),
where =˙
QQ:R→L(Vφ∗)is called the spin tensor of the frame φ∗relative to the inertial frame φ0.
Note that the inertial force vanishes if the change of frame φ0→φ∗is a Galilean transformation, i.e.,
˙
Q=0and¨
c∗=0, and hence, the equation of motion in the frame φ∗also takes the simple form
m∗¨
x∗=f∗,
which implies that the frame φ∗is also an inertial frame.
Therefore, any frame of reference obtained from a Galilean change of frame from an inertial frame is also
an inertial frame. Hence, all inertial frames form an equivalent class G, such that for any φ,φ∗∈G, the change
of frame φ→φ∗is a Galilean transformation. The Galilean class Gis a subclass of the Euclidean class E.
Material frame-indifference controversy 7
Remark 3 Since Euclidean change of frame is an equivalence relation, it decomposes all frames of reference
into disjoint equivalence classes, i.e., Euclidean classes as we previously called. However, the existence of an
inertial frame which is essential in establishing dynamic laws in mechanics leads to a special choice of the
Euclidean class of interest.
Let Ebe the Euclidean class which contains an inertial frame. Since different Euclidean classes are not
related by any Euclidean transformation,nor by any Galilean transformation, hence it is obvious that the Euclid-
ean class Eis the only class containing the subclass Gof all inertial frames. Consequently, from now on, the
only Euclidean class of interest for further discussions is the one denoted by E, containing the Galilean class
of all inertial frames.
In short, we can assert that physical laws, like the equation of motion, are in general not (Euclidean)
frame-indifferent. Nevertheless, the equation of motion is Galilean frame-indifferent, under the assumption
that mass and force are frame-indifferent quantities. This is usually referred to as Galilean invariance of the
equation of motion.
Motivated by classical mechanics, the balance laws of mass, linear momentum and energy for deformable
bodies
˙ρ+ρdiv ˙
x=0,
ρ¨
x−div T=ρb,
ρ˙ε+div q−T·grad ˙
x=ρr,(8)
in an inertial frame are required to be invariant under Galilean transformation. Since two inertial frames are
related by a Galilean transformation, it means that Eqs. (8) should hold in the same form in any inertial frame.
In particular, the balance of linear momentum takes the forms in the inertial frames φ,φ∗∈G,
ρ¨
x−div T=ρb,ρ
∗¨
x∗−(div T)∗=ρ∗b∗.
Since the acceleration ¨
xis Galilean objective, in order this to hold, it is usually assumed that the mass density ρ,
the Cauchy stress tensor Tand the body force bare objective scalar, tensor and vector quantities, respectively.
Similarly, for the energy equation, it is also assumed that the internal energy εand the energy supply rare
objective scalars, and the heat flux qis an objective vector. These assumptions concern the non-kinematic
quantities, including external supplies (b,r)and the constitutive quantities (T,q,ε).
In fact, for Galilean invariance of the balance laws, only frame-indifference with respect to Galilean trans-
formation for all those non-kinematic quantities would be sufficient. However, similar to classical mechanics,
it is postulated that they are not only Galilean objective but also Euclidean objective. Therefore, with the known
transformation properties of the kinematic variables, the balance laws in any arbitrary frame can be deduced.
To emphasize the importance of the objectivity postulate for constitutive theories, it will be referred to as
Euclidean objectivity for constitutive quantities:
Euclidean objectivity. The constitutive quantities: the Cauchy stress T, the heat flux qand the internal
energy density εare Euclidean objective (Euclidean frame-indifferent),
T(φ∗)=Q(t)T(φ ) Q(t),q(φ∗)=Q(t)q(φ), ε(φ∗)=ε(φ), (9)
where Q(t)∈O(Vφ,Vφ∗)is the isometric transformation of the change of frame from φto φ∗.
Note that this postulate concerns only frame-indifference properties of balance laws, so that it is a universal
property for any deformable bodies and therefore does not concern any aspects of material properties of the
body.
6 Constitutive equations in material description
Physically, a state of thermomechanical behavior of a body is characterized by a description of the fields of
density ρ(p,t), motion χ(p,t), and temperature θ(p,t). The material properties of a body generally depend
on the history of its thermomechanical behavior.
Let us introduce the notion of the history of a function. Let h(·)be a function of time. The history of hup
to time tis defined by
ht(s)=h(t−s),
8I.-S. Liu, R. Sampaio
where s∈[0,∞)denotes the time-coordinate pointing into the past from the present time t. Clearly, s=0
corresponds to the present time, therefore ht(0)=h(t).
Mathematical descriptions of material properties are called constitutive equations. We postulate that the
history of thermomechanical behavior up to the present time determines the properties of the material body.
Principle of determinism.Letφbe a frame of reference, and Cbe a constitutive quantity, then the con-
stitutive equation for Cis given by a functional of the form
C(φ, p,t)=Fφ(ρt,χt,θt;p), p∈B,t∈R,(10)
where the first three arguments are history functions:
ρt:B×[0,∞)→R,χt:B×[0,∞)→Eφ,θ
t:B×[0,∞)→R.
We call Fφthe constitutive function of Cin the frame φ. Such a functional allows the description of non-
local effect of an inhomogeneous body with memory of thermomechanical history. With the notation Fφ,we
emphasize that the value of a constitutive function depends on the frame of reference φin general.
For simplicity in further discussions on constitutive equations, we shall restrict our attention to material
models for mechanical theory only, and only the constitutive equation for the stress tensor will be considered.
General results can be found elsewhere ([3,6]).
In order to avoid possible confusions arisen from the viewpoint of employing different Euclidean spaces,
we shall be more careful about expressing relevant physical quantities in the proper space.
Let the set of history functions on a set Xin some space Wbe denoted by
H(X,W)={ht:X×[0,∞)→W}.
Then, the constitutive equation for the stress tensor, T(φ , p,t)∈Vφ⊗Vφ, can be written as
T(φ, p,t)=Fφ(χt;p), φ ∈E,p∈B,χt∈H(B,Eφ). (11)
Condition of Euclidean objectivity
Let φ∗∈Ebe another frame of reference, then the constitutive equation for the stress, T(φ∗,p,t∗)∈
Vφ∗⊗Vφ∗, can be written as
T(φ∗,p,t∗)=Fφ∗(∗(χt);p), p∈B,∗(χt)∈H(B,Eφ∗),
where the corresponding histories of motion are related by
∗(χt)( ¯p,s)=Qt(s)χt(¯p,s)−xo+c∗t(s),
for any s∈[0,∞)and any ¯p∈B, in the change of frame φ→φ∗givenby(1).
We need to bear in mind that according to the assumption referred to as the Euclidean objectivity (9), the
stress is a frame-indifferent quantity under a change of observer,
T(φ∗,p,t∗)=Q(t)T(φ , p,t))Q(t).
Therefore, it follows immediately that
Fφ∗(∗(χt);p)=Q(t)Fφ(χt;p)Q(t),χt∈H(B,Eφ), (12)
where Q(t)∈O(Vφ,Vφ∗)is the isometric transformation of the change of frame ∗:φ→φ∗.
The relation (12) will be referred to as the condition of Euclidean objectivity. It is a relation between the
constitutive functions relative to two different observers. In other words, different observers cannot indepen-
dently propose their own constitutive equations. Instead, the condition of Euclidean objectivity (12) determines
the constitutive function Fφ∗once the constitutive function Fφis given or vice versa. They determine one from
the other in a frame-dependent manner.
Material frame-indifference controversy 9
7 Principle of material frame-indifference
It is obvious that not any proposed constitutive equations would physically make sense as material models. First
of all, they may be frame-dependent. However, since the constitutive functions must characterize the intrinsic
properties of the material body itself, it should be observer-independence in certain sense. Consequently, there
must be some restrictions imposed on the constitutive functions so that they would be indifferent to the change
of frame. This is the essential idea of the principle of material frame-indifference.
Remark 4 In the case that does not distinguish the Euclidean spaces relative to different observers, i.e., Eφ=
Eφ∗=Eand Vφ=Vφ∗=V, as adopted usually in the literature ([3,6,17]), the principle of material
frame-indifference can simply be postulated as
Fφ(•;p)=Fφ∗(•;p), p∈B(13)
where •represents any history of motion in H(B,E)which is the common domain of the two functionals, and
their values are in the same tensor space V⊗V.
This states that for different observers φ,φ∗∈E, they all have the same constitutive function, Fφ=Fφ∗.
Note that in (13) the material point pis superfluously indicated to emphasize that it is valid only when the
material description is used, because in referential description, the choice of reference configuration may
change material properties. For example, it is obvious that after a shear deformation an isotropic material is
no longer isotropic. Therefore, if such a deformed state is chosen as a reference configuration, the material
property would have changed accordingly.
Recall that a change of frame ∗:φ→φ∗is associated with an isometry between Eφand Eφ∗,and
conversely, given an isometry
i:Eφ→Eφ∗,i(x)=I(x−xi)+x∗
i,(14)
there is a change of frame φ→φitaking (x,t)→ (i(x), t). For this change of frame, I∈O(Vφ,Vφ∗)and
Eφi=Eφ∗. Therefore, the following condition of Euclidean objectivity (12) must hold,
Fφi(i(χt);p)=I(t)Fφ(χt;p)I(t),χt∈H(B,Eφ), (15)
for which i(χt)∈H(B,Eφ∗), and the values of Fφi∈Vφ∗⊗Vφ∗.
Now, consider another isometry between Eφand Eφ∗,
j:Eφ→Eφ∗,j(x)=J(x−xj)+x∗
j.(16)
Similarly, there is a change of frame φ→φjwith J∈O(Vφ,Vφ∗)taking (x,t)→ (j(x), t),andEφj=Eφ∗,
and we have the condition of Euclidean objectivity,
Fφj(j(χt);p)=J(t)Fφ(χt;p)J(t),χt∈H(B,Eφ), (17)
for which j(χt)∈H(B,Eφ∗), and the values of Fφj∈Vφ∗⊗Vφ∗as before.
Note that the two isometries induce two changes of frame from φto two different frames φiand φjin the
same Euclidean space Eφ∗. Moreover, the two constitutive functions Fφiand Fφjhave the common domain
H(B,Eφ∗)and their values are in the same space Vφ∗⊗Vφ∗. Therefore, we can postulate:
Principle of material frame-indifference.Letφ→φiand φ→φjbe two changes of frame induced
by two isometries i,j:Eφ→Eφ∗, then the corresponding constitutive functions Fφiand Fφjmust have the
same form
Fφi(•;p)=Fφj(•;p), p∈B,•∈H(B,Eφ∗). (18)
This simple relation renders mathematically the basic idea of frame-indifference of material behavior: the
constitutive function, which models the intrinsic behavior of the material, is independent of observer, i.e.,
Fφi=Fφj(equivalence of (13)whenEφ=Eφ∗).
10 I.-S. Liu, R. Sampaio
We can now easily deduce the restriction on the constitutive function imposed by the principle of material
frame-indifference. From the relation (15)and(17), we have
Fφ(χt;p)=I(t)Fφi(i(χt);p)I(t)
=I(t)Fφj(i(χt);p)I(t)
=I(t)Fφj(j◦(j−1◦i)(χt);p)I(t)
=I(t)J(t)Fφ(( j−1◦i)(χt);p)J(t)I(t).
The underlined composite mapping q=(j−1◦i)is a Euclidean transformation (an isometry) from Eφto
itself,
q:Eφ→Eφ,q(x)=Q(t)(x−x0)+r(t), (19)
where Q=JI∈O(Vφ)is an orthogonal transformation on Vφ,andsomex0,r(t)∈Eφ. Indeed, from
(16), we have for any x∗∈Eφ∗,
j−1(x∗)=x=J(x∗−x∗
j)+xj,
and hence with (14), the transformation (19) follows from
(j−1◦i)(x)=JI(x−x0)+JI(x0−xi)+J(x∗
i−x∗
j)+xj,
with r(t)being the terms in the big parentheses.
Therefore, from the above relations, we obtain the following consequence of the principle of material
frame-indifference:
Condition of material objectivity. In a frame of reference φ, the constitutive function Fφmust satisfy
the condition
Fφ(q(χt);p)=Q(t)Fφ(χt;p)Q(t),p∈B,(20)
for any history of motion χt∈H(B,Eφ)and any Euclidean transformation
q:Eφ→Eφ,q(x)=Q(t)(x−x0)+r(t),
with some orthogonal transformation Q(t)∈O(Vφ),andsomex0,r(t)∈Eφ.
Since the condition (20) involves only one single frame of reference φ, it imposes a restriction on the
constitutive function Fφ. Sometimes, the condition of material objectivity is referred to as the “principle of
material objectivity”, to impart its relevance in characterizing material property and Euclidean objectivity, as
a more explicit form of the principle of material frame-indifference. Indeed, the original principle of material
frame-indifference in the fundamental treatise by Truesdell and Noll [17] was formulated in the form (20)
instead of the more intuitive expression (18).
Remark 5 There is an apparent similarity between the two relations (12)and(20). We emphasize that in the
condition of Euclidean objectivity (12), Q(t)is the orthogonal (isometric) transformation associated with the
change of frame from φto φ∗,while the condition of material objectivity (20) is valid in a single frame φ
for some arbitrary orthogonal transformation Q(t). Nevertheless, this apparent similarity still causes some
confusions in the literature in the occasional use of the “principle of frame-indifference” ([2,16,18]). From
our discussions so far, as we understand, the frame-indifference should not be regarded as a physical princi-
ple, it merely concerns the transformation properties due to changes of frame for kinematic or non-kinematic
quantities, while the principle of material frame-indifference concerns material properties relative to different
observers.
Material frame-indifference controversy 11
8 Constitutive equations in referential description
For mathematical analysis, it is more convenient to use referential description so that motions can be defined on
the Euclidean space instead of the set of material points. Therefore, for further discussions, we shall reinterpret
the principle of material frame-indifference for constitutive equations, or equivalently the condition of material
objectivity, relative to a reference configuration.
Let κ:B→Wt0be a reference placement of the body at some instant t0(see Fig. 2), then κφ=φt0◦κ:
B→Eφis the reference configuration of Bin the frame φ,and
X=κφ(p)∈Eφ,p∈B,Bκ=κφ(B)⊂Eφ.
The motion χ:B×R→Eφrelative to the reference configuration κφis given by
χκ(·,t):Bκ→Eφ,x=χ(p,t)=χκ−1
φ(X), t=χκ(X,t), χ=χκ◦κφ.
We can define the corresponding constitutive functions with respect to the reference configuration,
Fφ(χt;p)=Fφ(χt
κ◦κφ;κ−1
φ(X)) := Fκ(χt
κ;X),
and from (20), the condition of material objectivity for the constitutive function in the reference configuration
can be restated as
Fκ(q(χt
κ);X)=Q(t)Fκ(χt
κ;X)Q(t),X∈Bκ,(21)
for any history of motion χt
κ∈H(Bκ,Eφ)and any Euclidean transformation
q:Eφ→Eφ,q(χκ(X,t)) =Q(t)(χκ(X,t)−x0)+r(t),
for some orthogonal transformation Q(t)∈O(Vφ),andsomex0,r(t)∈Eφ.
Remark 6 Note that the condition (21) is valid for any Euclidean transformation q:Eφ→Eφ, which can also
be interpreted as a time-dependent rigid deformation of the body in the Euclidean space Eφ. This interpretation
is sometimes viewed as an alternative version of the principle of material frame-indifference and is called the
“principle of invariance under superimposed rigid body motions”.
9 Simple materials
According to the principle of determinism (10), thermomechanical histories of any part of the body can affect
the response at any point of the body. In most applications, such a non-local property is irrelevant. Therefore,
it is usually assumed that only thermomechanical histories in an arbitrary small neighborhood of Xaffect the
material response at the point X, and hence, the global history functions can be approximated at Xby Taylor
series up to a certain order in a small neighborhood of X. In particular, when only linear approximation is
concerned, the constitutive function is restricted to a special class of materials,
Fκ(χt
κ(·), X)=Hκ(∇XX
Xχt
κ(X), X),
so that we can write the constitutive equation for the stress as
T(X,t)=Hκ(Ft
κ;X), Ft
κ∈H({X},L(Vφ)), X∈Bκ,(22)
where Ft
κ=∇
XX
Xχt
κis the deformation gradient and the domain of the history is a single point {X}. Note that
although the constitutive function depends only on local values at the position X, it is still general enough
to define a material with memory of local deformation in the past. A material with constitutive equation (22)
is called a simple material by Noll. The class of simple materials is general enough to include most of the
materials of practical interests, such as elastic solids, viscoelastic solids, as well as Navier-Stokes fluids and
non-Newtonian fluids.
For the Euclidean transformation q:Eφ→Eφand x=χκ(X,t)from (21), we have
∇XX
Xq(χt
κ(X)) =∇
xx
xq(χt
κ(X))∇XX
Xχt
κ(X)=QtFt
κ(X).
12 I.-S. Liu, R. Sampaio
Therefore, we obtain the following main result for simple materials:
Condition of material objectivity. For simple materials relative to a reference configuration, the consti-
tutive equation T(X,t)=Hκ(Ft
κ;X)satisfies
HκQtFt
κ;X=Q(t)Hκ(Ft
κ;X)Q(t),(23)
for any history of deformation gradient Ft
κ∈H({X},L(Vφ)) and any orthogonal transformation Q(t)∈
O(Vφ).
Remark 7 The condition (23) is the most well-known result in constitutive theories of continuum mechanics.
It is the ultimate goal to obtain this result regardless of whoever agree or disagree with each other on the
formulation and interpretation of frame-indifference and the principle of material frame-indifference.
It is also interesting to see how the principle of material frame-indifference in the form (18), i.e., Fφi=Fφj,
reals in referential description. Let κ:B→Wt0be a reference placement of the body at some instant t0,and
i:Eφ→Eφ∗be an isometry, then
κi:B→Eφ∗,κ
i=i◦κφ=i◦φt0◦κ
is the reference configuration of Bin the frame φi=i◦φt0under the isometry i,and
Xi=κi(p)∈Eφ∗,p∈B,Bκi=κi(B)⊂Eφ∗.
The motion χ:B×R→Eφrelative to the reference configuration κiis given by
χκi(·,t):Bκi→Eφ∗,xi=χφi(p,t)=i(χ(p,t)) =χκi(κi(p), t),
so that we have
χφi=i◦χ=χκi◦κi,χκi=i◦χ◦κ−1
i.(24)
We can define the constitutive functions with respect to the reference configuration κi,
Fφi(i(χt);p)=Fφi(χt
κi◦κi;κ−1
i(Xi)) := Fκi(χt
κi;Xi),
and we can do similar things for another isometry j:Eφ→Eφ∗. Then, from (18), after some calculations,
we obtain
Fκi(χt
κj◦q∗;Xi)=Fκj(χt
κj;Xj), (25)
for any deformation history χt
κj∈H(Bκj,Eφ∗)and the isometry q∗=j◦i−1:Eφ∗→Eφ∗with its associated
orthogonal transformation Q∗=JI
∈O(Vφ∗).
Moreover, from (15)and(17), we obtain by the use of verifiable relations, j◦i−1=κj◦κ−1
iand
χκj=q∗χκiq∗−1(from (24)),
Fκi(q∗(χt
κi);Xi)=Q∗(t)Fκi(χt
κi;Xi)Q∗(t),Xi∈Bκi,(26)
for any deformation history χκi∈H(Bκi,Eφ∗)and any Euclidean transformation q∗:Eφ∗→Eφ∗with some
orthogonal transformation Q∗(t)∈O(Vφ∗).
Of course, we can see that the above condition (26) formulated in the Euclidean space Eφ∗is equivalent to
the condition of material objectivity (21) derived directly from (20)inEφ.
Remark 8 In the principle of material frame-indifference, we have postulated that constitutive functions are
independent of observers stated as Fφi=Fφjin (18). We emphasize that this is valid only when it is formu-
lated in material description. Indeed, from the above discussion, we have Fκi= Fκjin referential description,
instead, they must satisfy the relation (25).
Material frame-indifference controversy 13
10 Some general remarks
Remark 9 It is interesting to give the following example, which shows typically why some misconception
persisted. Let Fκ,F∗
κand T=Tκ(Fκ), T∗=T∗
κ(F∗
κ)be the deformation gradients and the constitutive
equations for the stress in two different frames relative to some reference configuration κ. One finds, in most
textbooks, that the objectivity conditions are given by
F∗
κ=QF
κ,T∗
κ(F∗
κ)=QTκ(Fκ)Q,
and the principle of material frame-indifference by
T∗
κ(•)=Tκ(•),
which combine to give the well-known condition of material objectivity,
Tκ(QF
κ)=QTκ(Fκ)Q.
This is the correct result equivalent to the condition (23). However, we have already shown that F∗
κ=QF
κis
valid only when the reference configuration is unaffected by the change of frame, and the principle of material
frame-indifference does not imply T∗
κ(•)=Tκ(•)in referential description. Nevertheless, the lucky incident
that the two inadequate assumptions would lead to the correct general result is quite striking and might have
contributed to some misconception over the decades.
Remark 10 To discuss the notion of invariance under observer changes in Sec. 20.1 of [2], it distinguishes the
observed space and the reference space, the former being the ambient space through which the body evolves
and the latter being the body at the reference configuration. And it emphasizes that
while a change of frame
affects the observed space through which the deformed body evolves, it does not affect the reference space
.In
order to emphasize this concept, the transformation of the relative deformation gradient under the change of
frame is given as an example. The relative deformation gradient,
F(t)(τ) =F(τ ) F(t)−1,
is derived using the configuration at time tas reference; hence, [it states that] F(t)is unaffected by the change
of frame. Therefore, the transformation law for F(t)(τ ) should be the same as that for F(τ ), namely (p. 150,
[2]),
F∗
(t)(τ ) =Q(τ) F(t)(τ ), (27)
instead of the usual one (see Sec. 29 of [17] and elsewhere),
F∗
(t)(τ ) =Q(τ) F(t)(τ )Q(t).(28)
To sho w t hat (27) is incorrect, let us consider simple materials. Since
F(X,t−s)=F(t)(X,t−s)F(X,t),
we can write the constitutive function in general as
T(X,t)=T(Ft(X,s)) := F(Ft
(t)(X,s), F(X,t)).
For simple fluids, one can show that the dependence of Freduces to the dependence of the density,1
T(X,t)=F(Ft
(t)(X,s), ρ(X,t)),
and the material objectivity condition,
F(Ft∗
(t),ρ) =Q(t)F(Ft
(t),ρ)Q(t),
1The properties of material symmetry have not been treated in this paper, because the concept of material symmetry trans-
formation concerns only changes of reference configuration in a fixed frame of reference; therefore, no changes of frame are
involved.
14 I.-S. Liu, R. Sampaio
can be written in the following form if the relation (27) is correct:
F(QtFt
(t),ρ) =Q(t)F(Ft
(t),ρ)Q(t).(29)
Since this condition must hold for any Q(t)∈O(V),letFt
(t)=Rt
(t)Ut
(t)be the polar decomposition, and with
Qt=(Rt
(t)), the condition (29) implies that
F(Ft
(t),ρ) =F(Ut
(t),ρ),
since Q(t)=(R(t)(t))=I. In other words, the dependence of the function Fon Ft
(t)reduces to only its
right stretch part Ut
(t).
Now return to the condition (29), for the argument on the left- hand side, since
QtFt
(t)=(QR
t
(t))Ut
(t),
which by uniqueness of polar decomposition, the right stretch part of QtFt
(t)is Ut
(t). Therefore, the condition
(29) becomes
F(Ut
(t),ρ) =Q(t)F(Ut
(t),ρ)Q(t)∀Q(t)∈O(V), (30)
which implies that FQ=QFfor any orthogonal transformation. Therefore, the constitutive function Fmust
reduce to a function proportional to the identity tensor.
This conclusion is obviously absurd for simple fluids
in general
. This proves that the relation (27) cannot be correct.
On the other hand, to get the correct result with the use of the correct transformation law (28), one can
prove, instead of (30), the following condition (see for example, [3,17]):
F(Q(t)Ut
(t)Q(t),ρ) =Q(t)F(Ut
(t),ρ)Q(t)∀Q(t)∈O(V),
for the reduced constitutive equation T(X,t)=F(Ut
(t)(X,s), ρ(X,t)) of simple fluids in general.
Remark 11 On frame-free theory:
Noll [13] pointed out that it should be possible to make the principle of material frame-indifference vacu-
ously satisfied by a frame-free formulation in describing internal interactions of a physical system. We would
liketotakealookatsuchatheory.
Setting aside the somehow esoteric mathematical structure and notations used by Noll (an interested reader
can consult [14]), we shall try to examine the intricate definitions for its underlaying assumptions, using our
conventional notations, hopefully without loss of their essential meanings.
First of all, we have to clarify some terminology. Without using a frame of reference, he defines a body
Bas a three-dimensional differentiable manifold, and a mapping κ:B→E, from the body into a Euclidean
space E, is called a placement (what we call configuration). The mapping γ◦κ−1, from one placement κto
another γ, is called transplacement (what we call deformation).
A placement κinduces a metric on the body, dκ(X,Y)=|κ(X)−κ(Y)|for all X,Y∈B. He calls this
metric the configuration induced by the placement κ(not to be confused with what we call configuration). By
this definition, any two placements differ by an isometric transplacement, such as a rigid body motion, will
give rise to the same configuration.
The intrinsic stress S is defined as a symmetric tensor on the tangent space of the body manifold. It is then
postulated that a frame-free constitutive law should involve only such intrinsic stresses [14], and for an elastic
material element, the response function of the intrinsic stress is defined in the following form (Eq. (2.6) of
[13]):
S=h(G), (31)
where G is called the configuration of the tangent space induced by the configuration dκof the placement
κ. Such a configuration is represented by the inner product induced by the positive-definite bilinear function
KKon the tangent space TXat X∈B,where K=∇
Xκ:TX→Vis the placement gradient, and Vis
the translation space of the Euclidean space E. (It is essentially the second-order tensor FFin our notation
with Fbeing the deformation gradient.) Therefore, given another placement ¯κ, such that ¯
K=∇
X¯κ=QK
Material frame-indifference controversy 15
for some Q∈Orth(V), it follows that ¯
K¯
K=KK, and hence it gives rise to the same configuration G.—In
other words, the constitutive equation (31) is invariant under superposed rigid body motions by definition.
The intrinsic stress is defined as S =F−1TF
−,whereTis the Cauchy stress. For elastic materials, it is
well known that the reduced constitutive equation, which satisfies the principle of material frame-indifference
identically, can be expressed in terms of the second Piola-Kirchhoff stress tensor, ˜
T=(det F)S, in the form
(it follows from the condition (23), also see [3,17], etc.),
˜
T=√det CS=t(C),
where C=FTFis the right Cauchy–Green strain tensor or essentially S =h(C). In other words, the
frame-free theory conveniently takes the well-known result from the principle of material frame-indifference
as axiomatic definition of intrinsic response function (31).
Surely, it is remarkable to have an interesting frame-free theory of Noll. However, from the above observa-
tions, the claim that such a formulation makes the principle of material frame-indifference vacuous is largely
misleading. Because it essentially employs the well-known consequences of the principle in its axiomatic
definition of configuration as the metric of placement and the response function for the conveniently chosen
intrinsic stress. Therefore, it seems that the premises of the frame-free theory of elasticity [13] are built on the
basic ideas of material frame-indifference (or equivalently, invariant under superimposed rigid body motions)
as the hidden foundation.
Remark 12 The first controversy on MFI was raised by Müller in 1972 [9] that the kinetic theory of gases does
not support the principle of material frame-indifference, according to which constitutive functions must be
frame-independent. In that article, an iterative scheme akin to the Maxwellian iteration from moment equations
in the kinetic theory, initiated with a certain equilibrium state, is employed to obtain the first and the second
iterates for stress and heat flux in terms of basic fields of continuum mechanics, namely fields of density,
motion and temperature. These relations are then regarded as constitutive equations. Indeed, the first iterates
yield Navier–Stokes and Fourier laws, and unfortunately (or surprisingly), the much elaborated second iterates
contain terms depending on the rotation of the frame—a clear violation of MFI in continuum mechanics. This
controversy set off spontaneous debates on the validity of MFI and all-round discussions on the concept of
frame-indifference ever since (see references in [1] and elsewhere).
In 1983, after the formulation of extended thermodynamics [7] for extended basic fields including stress
and heat flux, in the framework of continuum mechanics (including MFI with Euclidean transformations),
Müller sees “the violation of MFI in a new light”. It only reflects the frame dependence of the basic equations
of balance (for stress and heat flux), while the theory is frame-independent in the constitutive relations [sic].
Indeed, this reminds us of the earlier discussion on Galilean invariance of Newton’s law in Sec. 5. If we
regard Newton’s second law as a constitutive equation for the force in terms of motion, then in a non-inertial
frame, it is frame-dependent containing the rotation of the frame, because Newton’s second law is Galilean
invariant only. However, this is not what we would have done, instead, the force is postulated as a frame-
indifferent vector quantity and the frame-dependent terms are recognized as inertial forces (Coriolis force,
for example) in a non-inertial frame. We can now draw the analogy with the balance equation for stress (in
kinetic theory or extended thermodynamics) that the balance equation for the stress is Galilean invariant only
and hence frame-dependent, while the stress is a frame-indifferent tensor quantity. Those controversial iterates
are merely approximations of the balance equation itself, therefore containing frame-dependent terms in a
non-inertial frame, which may contain contribution from Coriolis force due to rotation of the frame. In other
words, regarding the Maxwellian iterates in the kinetic theory of gases as constitutive equations in the sense
of continuum mechanics, instead of as approximations of higher-order moment equations of balance near a
certain equilibrium state, is the source of confusion in this controversy.
Appendix: Isometries of Euclidean spaces
For a Euclidean space E, there is a vector space V, called the translation space of E, such that the difference
v=x2−x1of any two points x1,x2∈Eis a vector in V. We also require that the vector space Vbe equipped
with an inner product (·,·)V, so that length and angle can be defined.
Let V∗be the translation space of another Euclidean space E∗and L(V,V∗)be the space of linear trans-
formations from Vto V∗.ForA∈L(V,V∗), the transpose (or more generally called adjoint) of Adenoted by
A∈L(V∗,V)satisfies (u,Av∗)V=(Au,v∗)V∗for u∈Vand v∗∈V∗.
16 I.-S. Liu, R. Sampaio
Remark In an inner product space, since the norm is defined as uV=(u,u)V, from the identity, u+
v2
V=u2
V+v2
V+2(u,v)V,it follows that
u2
V=u∗2
V∗⇐⇒ (u,v)V=(u∗,v∗)V∗,
for any corresponding u,v∈Vand u∗,v∗∈V∗.
Definition I∈L(V,V∗)is called an isometric transformation if IuV∗=uVfor any u∈V.Let
O(V,V∗)denote the set of all isometric transformations in L(V,V∗).
From the above observation, an isometric transformation preserves the norm as well as the inner product,
the length and the angle.
Definition (Isometry): A bijective map i:E→E∗is an isometry if for x∈E,
x∗=i(x)=I(x−x0)+x∗
0,(32)
for some x0∈E,x∗
0∈E∗and some I∈O(V,V∗).
Let L(V)=L(V,V)be the space of linear transformations and O(V)=O(V,V)be the group of orthog-
onal transformation on V. Note that O(V,V∗)does not have a group structure in general. The transformation
(32) is often referred to as a Euclidean transformation when E=E∗and V=V∗. In this case, I∈O(V)is
an orthogonal transformation.
For I∈O(V,V∗), it follows that II=IVand II
=IV∗are identity transformations. Hence,
I=I−1and I∈O(V∗,V)is an isometric transformation from V∗to V. Moreover, if R∈O(V,V∗),
then IR∈O(V)and IR
∈O(V∗)are orthogonal transformations on Vand V∗, respectively. Indeed,
isometric transformation is the counterpart of orthogonal transformation when two different vector spaces are
involved.
Note that for clarity we have denoted transformations within the same space by capital letters, like Q∈
O(V)⊂L(V), and transformations between two different spaces by script capital letters, like Q∈O(V,V∗)⊂
L(V,V∗).
Examples in R2
Let E=R2and V=R2be its translation space with the standard inner product
(u,v)V=u1v1+u2v2,for u=(u1,u2), v=(v1,v
2),
and let E∗=R2and V∗=R2be its translation space with the inner product given by
(u∗,v∗)V∗=a2u∗
1v∗
1+b2u∗
2v∗
2,for u∗=(u∗
1,u∗
2), v∗=(v∗
1,v
∗
2),
for some constant a>0andb>0, which means it has different scaling in x-andy-directions.
Let the linear transformations Rand R∗be given in matrix form relative to the standard basis β=
{(1,0), (0,1)}of R2by
[R]=cos θsin θ
−sin θcos θ,[R∗]=⎡
⎣
cos θb
asin θ
−a
bsin θcos θ⎤
⎦.(33)
Obviously, the transformation Ris orthogonal on V,R∈O(V), while one can show that R∗is orthogonal on
V∗,R∗∈O(V∗).Indeed,foru∗=(x∗,y∗)and ˆ
u∗=R∗u∗=(ˆx∗,ˆy∗),wehave
ˆx∗=x∗cos θ+b
ay∗sin θ, ˆy∗=−a
bx∗sin θ+y∗cos θ,
and hence
R∗u∗V∗=a2(ˆx∗)2+b2(ˆy∗)2=a2(x∗)2+b2(x∗)2=u∗V∗.
Material frame-indifference controversy 17
Fig. 4 The effect of scaling and rotation: the views of the observer in space Eon the left and in space E∗on the right. In this
example, θ=tan−1(1/2)and a=2/3,b=1
Now consider a linear transformation S:V→V∗given by
[S]=1/a0
01/b,(34)
in matrix form relative to the stand basis. One can easily show that it is an isometric transformation, S∈
O(V,V∗), i.e., SuV∗=uV.
Geometrically, this transformation maps a figure in Vto a similar figure scaled down/up in two different
directions (xand y)inV∗. This is like drawings of floor plan or street maps, in which the corresponding
segments have the same physical length but in different scale, or in two different snapshots of the same figure
from different distances.
We can also consider the composite maps of (33)and(34), and, by direct computation, show that
Q=SR=R∗S,
which is a linear transformation from Vto V∗givenby(seeFig.4)
[Q]=⎡
⎢
⎣
1
acos θ1
asin θ
−1
bsin θ1
bcos θ
⎤
⎥
⎦.
One can show that Q∈O(V,V∗)is an isometric transformation, i.e., QuV∗=uV.
In Fig. 4, for an illustration, we show the image of a square as seen by a viewer in the space E
and its image as seen by another viewer in the space E∗. On the left-hand side, the E-viewer is seeing
the square in a screen with aspect ratio of 4:3 as in standard television, while on the right-hand side,
the E∗-viewer is seeing the stretched object in the wide-screen with aspect ratio 6:3. Note that on the
top, the square is stretched into a rectangle, while at the bottom, the image of the rotated square is not
even a rectangle. As a little surprise, the transformation R∗looks like a rotation and a shearing, even
though we have shown that R∗is an orthogonal transformation in O(V∗)due to the uneven scaling in
the definition of inner product of V∗. Similarly, the isometric mapping Qtransforms a square in Vinto a
tilted parallelogram in V∗after a rotation Rfollowed by a stretching Sor a stretching Sfollowed by a
rotation R∗.
In analogy with the above example, in Fig. 5by switching from the standard 4:3 screen to
the wide-screen mode in a television, one may see the change in images that not only stretches
the human face horizontally, but also distorts the face by shearing when the head is tilted. Nev-
ertheless, in reality, both images belong to the same face as seen from different modes (viewing
spaces).
18 I.-S. Liu, R. Sampaio
Fig. 5 What one sees when switching from the standard 4:3 screen to the wide-screen mode of a television
Therefore, in consideration of change of observer, besides relative motion, it would be meaningful to allow
relative orientation as well as scaling between observers, i.e., consenting observers may belong to different
Euclidean spaces, related by isometric transformations.
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