Tensors: Lorentz vs Galilean invariance

[cesiumfrog]

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)?

 

[pmb_phy]

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

 

[Entropia]

Tensors play a crucial role in both Lorentz and Galilean invariance, but their significance differs in each case. In Lorentz invariance, tensors are used to describe physical quantities in a way that is independent of the observer's frame of reference. This means that the equations and laws of physics can be expressed in terms of tensors, and they will hold true for all observers, regardless of their relative motion. This is because the transformation rules for tensors in Lorentz invariance are based on the principles of special relativity, which state that the laws of physics should be the same for all inertial observers.

On the other hand, in Galilean invariance, tensors are used to describe physical quantities in a way that is independent of the observer's position and orientation in space. This means that the equations and laws of physics can be expressed in terms of tensors, and they will hold true for all observers, regardless of their location or orientation. This is because the transformation rules for tensors in Galilean invariance are based on the principles of classical mechanics, which state that the laws of physics should be the same for all observers in a non-accelerating frame of reference.

The physical significance of tensors lies in their ability to accurately describe and predict the behavior of physical systems, regardless of the observer's frame of reference. They allow us to understand and describe the fundamental laws of nature in a way that is consistent and universal, making them a powerful tool in both relativity theory and classical mechanics. By using tensors, we can ensure that our equations and laws of physics are valid for all observers, and this is essential in understanding the fundamental nature of our universe.