Acceleration Tensor - Rotating Frame

In summary: In SR, the metric is flat; in GR, the metric is curved. The acceleration tensor in GR is the Riemann curvature tensor, which is rank 4, not rank 1. Therefore, in summary, the acceleration of a rotating coordinate system does not transform as a rank 1 tensor, but rather as a rank 4 tensor in GR. This can be seen by considering the ill-behaved metric coefficients and the fact that the rotating coordinate system requires a non-Minkowskian metric.
  • #1
PrinceOfDarkness
31
0
If a coordinate system is rotating, that is time 't' is not independent, then does the acceleration transform as rank 1 tensor?

I thought that it wouldn't because when time is changing, so acceleration will change in a more complicated way than a rank 1 tensor. Perhaps as a rank 2 tensor.

This Q is really troubling me. There are two groups in my class, one saying it still transforms as a rank 1 tensor, the other saying it doesn't transform as a rank 1 tensor. Some even say that acceleration never transforms like a rank 1 tensor. I wonder how! I think it transforms like a rank 1 tensor if it goes fixed rotation, but 'rotating' coordinate system will mean that transformation is more complicated.
 
Physics news on Phys.org
  • #2
Rotations (with presumably fixed angular velocity) are examples of (restricted) Lorentz transformations. If you know how the acceleration behaves when being subject to a (restricted) Lorentz transformation, then everything would be fine, wouldn't you say...?

Daniel.
 
  • #3
PrinceOfDarkness said:
If a coordinate system is rotating, that is time 't' is not independent, then does the acceleration transform as rank 1 tensor?

You need to be a bit more specific here. The 4-acceleration is a geometric object, so it transforms as a tensor.

A rotating coordinate system will require a metric that is not Minkowskian, so you start getting into GR rather than SR.

The rotating coordinate system will be ill-behaved when r*w = c, some of the metric coefficients go to zero (or was it infinity? I'd have to double check - but I know they are not well-behaved).
 
  • #4
A rotating coordinate system will require a metric that is not Minkowskian, so you start getting into GR rather than SR.
No, you're merely getting into differential geometry.
 

Related to Acceleration Tensor - Rotating Frame

1. What is an acceleration tensor in a rotating frame?

An acceleration tensor in a rotating frame is a mathematical representation of the rate of change of acceleration with respect to time and position in a rotating coordinate system. It takes into account both the acceleration due to the motion of the frame and the acceleration of a body within the frame.

2. How is an acceleration tensor calculated?

An acceleration tensor is calculated using the laws of vector calculus and rotational dynamics. It involves taking the second derivative of the position vector in the rotating frame and applying the Coriolis and centrifugal forces.

3. What is the significance of an acceleration tensor in physics?

An acceleration tensor plays a crucial role in understanding the dynamics of objects in a rotating frame. It allows us to accurately calculate and predict the acceleration of objects in a non-inertial reference frame, which is important in fields such as aerospace engineering and astrophysics.

4. How does the acceleration tensor differ from the acceleration vector?

The acceleration tensor is a 3x3 matrix, while the acceleration vector is a single vector. The acceleration tensor takes into account the direction and magnitude of acceleration in all three dimensions, while the acceleration vector only represents the magnitude and direction of acceleration in one direction.

5. Can an acceleration tensor be used in a stationary frame?

Yes, an acceleration tensor can be used in a stationary frame as well. In this case, the acceleration tensor will have zero values for the Coriolis and centrifugal terms, as there is no rotation in a stationary frame. However, it can still be a useful tool for analyzing the acceleration of objects in a non-inertial reference frame.

Similar threads

I Interpreting The Definition of Tensors
    • Differential Geometry
Replies
6
Views
2K
A Rich "isotropic tensor" concept
    • General Math
Replies
1
Views
1K
I How to derive irreducible representations/tensors?
    • Special and General Relativity
Replies
22
Views
2K
I Basic Q about Vector/tensor definition and velocity
    • Differential Geometry
Replies
11
Views
2K
Vector fields, flows and tensor fields
    • Differential Geometry
Replies
33
Views
5K
I Connection b/w curvature and gravitational field-removing coordinate transformations
    • Special and General Relativity
2
Replies
40
Views
2K
Classical Books about tensor analysis just good enough for physics
    • Science and Math Textbooks
Replies
5
Views
2K
I Covariant divergence of vector; physical meaning with contracted Tuv
    • Special and General Relativity
Replies
2
Views
1K
I Exploring Tensors in Physics: Understanding Rank and When to Use Matrices
    • Other Physics Topics
Replies
2
Views
1K
I Meaning of the invariants built from the angular momentum tensor
    • Special and General Relativity
Replies
1
Views
172

Hot Threads

Recent Insights

Back
Top