Objectivity

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Objectivity the role of space-time models
Peter Ván HAS, RIPNP, Department of Theoretical
Physics

• Introduction objectivity
• Traditional objectivity - problems
• We need 4 dimensions
• Non-relativistic space-time model
• Some consequences
• Discussion

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Why nonequilibrium thermodynamics?

• Thermodynamics

science of temperature
Thermodynamics science of macroscopic
energy changes
general framework of any Thermodynamics
(?) macroscopic (?) continuum (?)
theories

• General framework
• Second Law
• fundamental balances
• objectivity - frame indifference

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Material frame indifference Noll (1958),
Truesdell and Noll (1965) Müller (1972, )
(kinetic theory) Edelen and McLennan (1973) Bampi
and Morro (1980) Ryskin (1985, ) Lebon and
Boukary (1988) Massoudi (2002) (multiphase
flow) Speziale (1981, , 1998),
(turbulence) Murdoch (1983, , 2005) and Liu
(2005) Muschik (1977, , 1998), Muschik and
Restuccia (2002) ..
Objectivity
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Nonlocalities
Requirements of objectivity ?
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Second Law
basic balances
(and more)

• basic state
• constitutive state
• constitutive functions

Second law
(universality)
Constitutive theory
Method Liu procedure
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Basic state space a (..)
Nonlocality in time (memory and inertia)
Nonlocality in space (structures)
Nonlocality in spacetime
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The principle of material frame-indifference
(material objectivity, form-invariance) The
material behaviour is independent of
observers. Its mathematical formulation The
material behaviour is described by a mathematical
relation having the same functional form for all
observers.
Mechanics Newton equation
frame
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What is a vector? element of a vector space -
mathematics something that transforms
according to some rules - physics (observer
changes, objectivity)
Rigid observers are distinguished
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Observers and reference frames
Noll (1958)
is a four dimensional objective vector, if
where
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Traditional objectivity
Vectors
Tensors
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Are there four vectors in non-relativistic
spacetime?
Motion
Velocity
definition
Traditionally non objective!
is an objective four vector
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Covariant derivatives as the spacetime is flat
there is a distinguished one.
covector field mixed tensor field
The coordinates of the covariant derivative of a
vector field do not equal the partial derivatives
of the vector field if the coordinatization is
not linear.
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(No Transcript)
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Material time derivative
Flow generated by a vector field V.
is the change of F along the integral curve.
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substantial time derivative
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Spec. 2 is a spacelike vector field
The material time derivative of a vector even
if it is spacelike is not given by the
substantial time derivative.
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Jaumann, upper convected, etc derivatives
In our formalism ad-hoc rules to eliminate the
Christoffel symbols. For example
upper convected (contravariant) time derivative
One can get similarly Jaumann, lower convected,
etc
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• Conclusions
• Objectivity has to be extended to a four
  dimensional setting.
• Four dimensional covariant differentiation is
  fundamental in non-relativistic spacetime. The
  essential part of the Christoffel symbol is the
  angular velocity of the observer.
• Partial derivatives are not objective. A number
  of problems arise from this fact.
• Material time derivative can be defined
  uniquely. Its expression is different for fields
  of different tensorial order.

space time ? spacetime
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Rotating observer is special there are
more. Observer and continuum is not the same
e.g. there are different angular velocities at
different points. We need a DEFINITION of the
observer and an observer independent formalism.
The clear and unquestionable principle of
material frame-indifference can be formulated
without referring to observers if we use a
convenient mathematical structure for
non-relativistic spacetime.
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What is non-relativistic space-time?
Absolute time.
Space-time M four dimensional affine space
(over the vector space M), Time I is a
one-dimensional affine space, Time evaluation ?
M?I is an affine surjection. Distance Euclid
ean structure on EKer(?)
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What is non-relativistic space-time?
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Space and time in space-time
A direction is necessary
E
M
x
0
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Consequences four vectors and covectors cannot
be identified, because there is not Euclidean
structure on M Differentiation of
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Fields
Derivatives
covector field mixed tensor field cotensor field
A tensor and cotensor fields do not have a
trace. A mixed tensor field does not have a
symmetric part.
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World line function
Velocity field
Mass-momentum balance
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Observers smooth a space point of an
observer is a curve in space-time
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Inertial observer Uconst.
M
U(x)
Ft(x)
x
t0 ?(x)
t
I
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Splitting of space-time
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Splitting of fields
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Relative form of absolute physical quantities
Scalar field
Vector field
Covector field
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Flow of a continuum The velocity field of a
continuum generates a flow, the map
Reference configuration current configuration
Relative form of the flow is the
motion
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Material time derivative
Scalar field
Space-like vector field
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Convected time derivatives VECTOR
Lie derivative of a space-like vector field
upper convected time derivative.
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Convected time derivatives COVECTOR
Space-like part of the Lie derivative of a
space-like covector field lower convected
time derivative.
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Discussion absolute Liu-procedure
(mechanics!) material frame
indifference the constitutive functions must
be absolute Traditional consequences of MFI
must be checked new models in rheology A
particular result If an absolute constitutive
function depends on , then the principle
of material frame-indifference does not exclude,
on the contrary, it requires that the angular
velocity of the observer appear explicitly in
the relative constitutive function.
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References Traditional Truesdell, C. and
Noll, W., The Non-Linear Field Theories of
Mechanics (Handbuch der Physik, III/3), Springer
Verlag, Berlin-Heidelberg-New York,
1965. Matolcsi, T. and Ván, P., Can material
time derivative be objective?, Physics Letters A,
2006, 353, p109-112, (math-ph/0510037). Space-tim
e models T. Matolcsi Spacetime Without
Reference Frames, Publishing House of the
Hungarian Academy of Sciences, Budapest ,
1993. Matolcsi, T. and Ván, P., Absolute time
derivatives 2006, (math-ph/0608065). Rheology B
ird, Byron R., Armstrong, R. C. and Hassager,
Ole Dynamics of polymeric liquids I., John Wiley
and Sons, Inc., New York-Santa Barbara-etc..,
1977.
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Thank you for your attention!