- #1
joda80
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Hi All,
I have a question about transformation matrices (sorry about the typo in the title). The background is that I've spent some time learning differential geometry in the context of continuum mechanics and general relativity, but I'm unable to connect some of the concepts.
So I have this book about general relativity (by T. Fliessbach), where it says that the transformation matrices of the local diffeomorphisms considered in general relativity are no tensors (justifying writing the indices above each other, rather than 'from northwest to southeast'). So far, so good.
But in continuum mechanics, there is a quantity called the "deformation-gradient tensor", which maps e.g., a material line segment from the initial configuration to a later configuration. Interpreting this map as a passive transformation (change of basis), this tensor is, in my understanding, precisely the transformation matrix from material coordinates to spatial coordinates (material coordinates are initially Cartesian but follow along with the flow and thus become curvilinear). So if the material coordinates are denoted by [itex]a^{\nu}[/itex] and the fixed spatial (Cartesian) coordinates by [itex]x^{i}[/itex], then a material line segment transforms like this:
[tex]d{x}^{i} =\frac{\partial x^{i}}{\partial a^{\nu}} da^{\nu},[/tex]
where the transformation matrix is just the deformation gradient.
The fact that in continuum mechanics, this general transformation matrix (from arbitrary curvilinear coordinates to Cartesian coordinates) is a tensor, but presumably not so in general relativity, confuses me.
In Marsden and Hughes' textbook (mathematical foundations of elasticity), it is stressed that the deformation gradient,
[tex]F^{i}_{\cdot \nu} = \frac{\partial \Phi^{i}}{\partial a^{\nu}}[/tex]
does not involve the covariant derivative because [itex]\Phi[/itex] is not a vector but a point mapping. Could this have something to do something with it?
I hope the problem is somewhat understandable.
Thanks in advance,
Johannes
I have a question about transformation matrices (sorry about the typo in the title). The background is that I've spent some time learning differential geometry in the context of continuum mechanics and general relativity, but I'm unable to connect some of the concepts.
So I have this book about general relativity (by T. Fliessbach), where it says that the transformation matrices of the local diffeomorphisms considered in general relativity are no tensors (justifying writing the indices above each other, rather than 'from northwest to southeast'). So far, so good.
But in continuum mechanics, there is a quantity called the "deformation-gradient tensor", which maps e.g., a material line segment from the initial configuration to a later configuration. Interpreting this map as a passive transformation (change of basis), this tensor is, in my understanding, precisely the transformation matrix from material coordinates to spatial coordinates (material coordinates are initially Cartesian but follow along with the flow and thus become curvilinear). So if the material coordinates are denoted by [itex]a^{\nu}[/itex] and the fixed spatial (Cartesian) coordinates by [itex]x^{i}[/itex], then a material line segment transforms like this:
[tex]d{x}^{i} =\frac{\partial x^{i}}{\partial a^{\nu}} da^{\nu},[/tex]
where the transformation matrix is just the deformation gradient.
The fact that in continuum mechanics, this general transformation matrix (from arbitrary curvilinear coordinates to Cartesian coordinates) is a tensor, but presumably not so in general relativity, confuses me.
In Marsden and Hughes' textbook (mathematical foundations of elasticity), it is stressed that the deformation gradient,
[tex]F^{i}_{\cdot \nu} = \frac{\partial \Phi^{i}}{\partial a^{\nu}}[/tex]
does not involve the covariant derivative because [itex]\Phi[/itex] is not a vector but a point mapping. Could this have something to do something with it?
I hope the problem is somewhat understandable.
Thanks in advance,
Johannes
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