Vector and Scalar Tensor Invariance

Click For Summary

Discussion Overview

The discussion revolves around the concept of tensor invariance, specifically in relation to velocity and energy. Participants explore the differences between scalar and vector quantities, and how these relate to different reference frames and coordinate systems, particularly under Galilean transformations.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant expresses confusion about tensor invariance, noting that while tensors should have the same value across coordinate systems, velocity is frame-dependent, leading to different kinetic energy measurements in different frames.
  • Another participant questions the type of coordinate systems being referenced, asking whether they are related by rotations, Galilean transformations, or Lorentz transformations.
  • A participant clarifies that under Galilean transformations, acceleration is invariant while velocity is not, suggesting that not all vectors are invariant under such transformations.
  • This participant explains that while a vector itself remains invariant under rotations, its components change according to a rotation matrix, emphasizing the distinction between the vector and its components.
  • The same participant elaborates on tensors of rank 2, stating that they have multiple components that also transform under rotations, and reiterates that scalars, as tensors of rank zero, maintain the same value across related coordinate systems.

Areas of Agreement / Disagreement

Participants demonstrate some agreement on the definitions and properties of tensors and vectors, but there remains a lack of consensus regarding the implications of tensor invariance in the context of frame-dependent quantities like velocity and energy.

Contextual Notes

The discussion highlights the complexity of tensor invariance and the specific conditions under which different quantities maintain their values across coordinate systems. There is an implicit assumption that the definitions of tensors and their transformations are understood, but the application to physical quantities like velocity and energy remains nuanced and unresolved.

e2m2a
Messages
354
Reaction score
13
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?
 
Physics news on Phys.org
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?
 
Galilean transformations.
 
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 = < Ax, Ay, Az> 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.
 
Yes it does. Thanks.
 

Similar threads

Undergrad Tensor Equations: Invariance Across Different Reference Frames?
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad Variant and Invariant Physical Quantities....
  • · Replies 10 ·
Replies
10
Views
5K
Undergrad Transformation Riemann tensor to an accelerating frame
  • · Replies 10 ·
Replies
10
Views
2K
Undergrad Invariance of a tensor of order 2
  • · Replies 2 ·
Replies
2
Views
3K
Undergrad What Are Higher Rank Tensors and When Should You Use Matrices in Physics?
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad How do Tensors "work" in relation to linear algebraic objects?
  • · Replies 7 ·
Replies
7
Views
1K
Undergrad How deep does it need to go for a physics theory to be consistent?
  • · Replies 50 ·
2
Replies
50
Views
5K
Undergrad Understanding Tensors & Knot Theory in Physics
  • · Replies 4 ·
Replies
4
Views
3K
Undergrad Meaning of the invariants built from the angular momentum tensor
  • · Replies 1 ·
Replies
1
Views
1K
Undergrad What is the relationship between the Hamiltonian and Lorentz invariance?
  • · Replies 6 ·
Replies
6
Views
3K