Vector and Scalar Tensor Invariance

[e2m2a]

I am confused about tensor invariance as it applies to velocity and energy. My understanding is a tensor is a mathematical quantity that has the same value for all coordinate systems. I also understand that a vector is a first order tensor and energy is a zero order tensor. Thus, they should have the same values for all coordinate systems.

However, velocity is a frame dependent quantity. One reference frame may measure the velocity of a particle to be 1 m/s, while another frame might measure the velocity of the same particle to be 10 m/s. Furthermore, if we assume the mass of the particle is 2 kg, then the first frame will measure a kinetic energy (scalar quantity) of 1 joule and the second frame will measure 100 joules.

Clearly, these tensor quantities are not invariant with respect to the two frames. Am I confusing coordinate systems with reference frames?

 

[Chandra Prayaga]

When you say, "all coordinate systems", are you talking about coordinate systems related to one another by rotations (or translations) in 3-dimensions, or by Galilean transformations (coordinate systems related by motion with constant velicities with respect to each other), or by Lorentz transformations, which include relativistic velocities?

 

[e2m2a]

Galilean transformations.

 

[Chandra Prayaga]

Under Galilean transformations, the acceleration of a particle is invariant, not the velocity, as you said. So it is certainly not true that every vector is invariant under a Galilean transformation.
I believe the concept that you are looking at is the following:
A vector in 3-dimensions is invariant under any rotations of the coordinate system.
Note that the vector is invariant, but not its components. The components transform under the rotation, according to standard rules. So if you have a vector A, it is written as
A = < A_x, A_y, A_z> in one coordinate system, with components as written inside the brackets. The same vector is written in another coordinate ayatem as:
A = < A'_x', A'_y', A'_z'> .
The components in one coordinate system are related to those in the other coordinate system through a rotation matrix.
Similarly, a tensor of rank 2 in 3-d is a quantity which has 9 components in a coordiante system. If you rotate the coordinate system, the same tensor will have 9 different components in the new coordinate system.
The components in one coordinate system are related to those in the other coordinate system through a rotation matrix. This is a 9 x 9 matrix.
A scalar, as you stated, is a tensor of rank zero. It has the same value (single component) in all coordinate systems related to each other by rotations in 3-d.
Hope this helps.

 

[e2m2a]

Yes it does. Thanks.