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

general framework of any Thermodynamics (?) macroscopic (?) continuum (?) theories Thermodynamics science of macroscopic energy changes Thermodynamics science of temperature Why nonequilibrium thermodynamics? General framework: – Second Law – fundamental balances – objectivity - frame indifference

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

Nonlocalities: Requirements of objectivity:?

Second Law: basic balances – basic state: – constitutive state: – constitutive functions: Second law: Constitutive theory Method: Liu procedure (universality) (and more)

Nonlocality in time (memory and inertia) Nonlocality in space (structures) constitutive space (weakly nonlocal) Nonlocality in spacetime Basic state space: a = (…..) ???

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

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: h (x,t) K K’K’

Observers and reference frames: is a four dimensional objective vector, if where Noll (1958)

Traditional objectivity: Vectors: Tensors:

Motion: is an objective four vector Are there four vectors in non-relativistic spacetime? definition: Traditionally non objective! Velocity:

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. If are inertial coordinates, the Christoffel symbol with respect to the coordinates has the form:

where is the angular velocity of the observer

Material time derivative: is the point at time t of the integral curve V passing through x. t0t0 t x F t (x) V(x) Flow generated by a vector field V. is the change of Φ along the integral curve.

t0t0 t x F t (x) V(x) is the covariant derivative of according to V. substantial time derivative Spec. 1: is a scalar

The material time derivative of a vector – even if it is spacelike – is not given by the substantial time derivative. Spec. 2: is a spacelike vector field

Jaumann, upper convected, etc… derivatives: In our formalism: ad-hoc rules to eliminate the Christoffel symbols. For example: for a spacelike vector upper convected (contravariant) time derivative One can get similarly Jaumann, lower convected, etc…

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

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. 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.

What is non-relativistic space-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:Euclidean structure on E=Ker(  ) Absolute time.

What is non-relativistic space-time? M=E  I M=EIM=EI E  I

Space and time in space-time x M E 0 A direction is necessary

Consequences: four vectors and covectors cannot be identified, because there is not Euclidean structure on M Differentiation of

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.

Velocity field: Mass-momentum balance: World line function:

t 0 =  (x) t x F t (x) U(x) M I Observers: smooth a space point of an observer is a curve in space-time

t 0 =  (x) t x F t (x) U(x) M I Inertial observer: U=const.

Splitting of space-time:

Splitting of fields:

Relative form of absolute physical quantities: Scalar field: Vector field: Covector field:

Flow of a continuum : The velocity field of a continuum generates a flow, the map motion Relative form of the flow is the: Reference configuration – current configuration

Material time derivative: Scalar field: Space-like vector field:

Convected time derivatives: VECTOR Lie derivative of a space-like vector field: upper convected time derivative.

Convected time derivatives: COVECTOR Space-like part of the Lie derivative of a space-like covector field: lower convected time derivative.

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.

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, Matolcsi, T. and Ván, P., Can material time derivative be objective?, Physics Letters A, 2006, 353, p , (math-ph/ ). Space-time models: T. Matolcsi: Spacetime Without Reference Frames, Publishing House of the Hungarian Academy of Sciences, Budapest, Matolcsi, T. and Ván, P., Absolute time derivatives 2006, (math-ph/ ). Rheology: Bird, 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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