Tensors: Lorentz vs Galilean invariance

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What is the physical significance of tensors?

Occasionally, motivating statements are made roughly along the lines of "if an equation can be expressed purely in terms of tensors, then it is true for all observers". But that doesn't seem quite complete because different tensor-users would have contradictory views on which coordinate frames do and do not represent physical observers.

As I understand it, the exact same set of simple transformation rules is used for translating the components of all tensors between all pairs of coordinate frames. What then distinguishes the tensor equations in relativity theory from tensor equations in Newtonian mechanics (or Engineering)?
 

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What is the physical significance of tensors?
There is no unique answer to that since different tensors apply to different physical phenomena. A tensor has a mathematical definition. The definition has to do with various things. E.g. a tensor called a "Cartesian Tensor" is a set of "component" which has a certain transformation property under an orthogonal transformation. A "Lorentz tensor" is a tensor which has a certain transformation property under a Lorentz transformation. etc.
Occasionally, motivating statements are made roughly along the lines of "if an equation can be expressed purely in terms of tensors, then it is true for all observers". But that doesn't seem quite complete because different tensor-users would have contradictory views on which coordinate frames do and do not represent physical observers.
That's a mouthful which I will respond to by citing the web pages I made for such an explanation. They are

http://www.geocities.com/physics_world/gr_ma/tensors_via_analytic.htm
http://www.geocities.com/physics_world/gr_ma/tensor_via_geometric.htm

What then distinguishes the tensor equations in relativity theory from tensor equations in Newtonian mechanics (or Engineering)?
The aloowable set of coordinate transformations. In the case of a Cartesian tensor, used in Newtonian physics, the allowable transformations are orthogonal transformations.

For a solid example of a tensor in Classical Physics please see

http://www.geocities.com/physics_world/mech/tidal_force_tensor.htm

Best wishes

Pete
 

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