# Maximal invariance group for constant acceleration?

• strangerep
In summary, Micromass contributed a post in the relativity forum discussing the most general coordinate transformation that preserves the property of zero acceleration. This is known as a fractional-linear transformation. There is also a well-developed body of mathematics, known as Galilean geometry, that covers transformations for curves with constant acceleration. This field has applications in both physics and mathematics.
strangerep
Recently, over in the relativity forum, Micromass contributed a post:
https://www.physicsforums.com/showpost.php?p=4168973&postcount=89
giving a proof that the most general coordinate transformation preserving the property of zero acceleration (i.e., maps straight lines to straight lines) is of fractional-linear form.

I'm now wondering whether there are generalizations of this sort of thing applicable to, e.g., trajectories of constant acceleration (i.e., nonzero, but constant acceleration), and maybe also trajectories of constant jerk?
[ http://en.wikipedia.org/wiki/Jerk_(physics) ]

1. What is the most general coordinate transformation (in 1+1D for starters) that maps curves of constant acceleration into curves of constant acceleration?

2. Is there already a well-developed body of math covering such cases, perhaps generalizing the theory of projective spaces that arise from an ambient linear space?

(P.S: I wasn't sure whether to ask this under Calculus, or Group Theory, or General Math. It kinda reaches into several categories.)

(P.P.S: Haven't tried using bold font in questions before. Thought I'd try it here since it seems to work for Tom Stoer... :-)

Last edited:

Hi there,

Thank you for bringing up this interesting topic! I am always fascinated by the connections between different areas of physics and mathematics.

To answer your first question, the most general coordinate transformation that maps curves of constant acceleration into curves of constant acceleration is known as a Galilean transformation. This transformation preserves the form of the equations of motion for objects moving with constant acceleration, meaning that the acceleration remains constant in all frames of reference. This is in contrast to the fractional-linear transformation mentioned in the original post, which preserves the property of zero acceleration.

As for your second question, there is indeed a well-developed body of mathematics that covers such cases. This area of mathematics is known as Galilean geometry, which generalizes the concepts of projective geometry to include the effects of constant acceleration. It is a fascinating field that has applications in both physics and mathematics.

I hope this answers your questions and provides some insight into the connections between different areas of physics and mathematics. Thank you again for bringing up this topic and sparking a discussion!

## 1. What is the maximal invariance group for constant acceleration?

The maximal invariance group for constant acceleration is a group of transformations that leave the equations of motion for a system with constant acceleration unchanged. In other words, these transformations do not affect the acceleration of the system.

## 2. How is the maximal invariance group for constant acceleration related to special relativity?

The maximal invariance group for constant acceleration is closely related to special relativity. In fact, it is a subgroup of the full Lorentz group, which is the symmetry group of special relativity. This means that the transformations in the maximal invariance group can be used to describe the behavior of systems with constant acceleration in the context of special relativity.

## 3. How is the maximal invariance group for constant acceleration connected to the concept of spacetime?

The maximal invariance group for constant acceleration is closely tied to the concept of spacetime. This group is a subgroup of the Poincaré group, which is the symmetry group of Minkowski spacetime. This means that the transformations in the maximal invariance group can be used to describe the behavior of systems with constant acceleration in the context of Minkowski spacetime.

## 4. What are some practical applications of the maximal invariance group for constant acceleration?

One practical application of the maximal invariance group for constant acceleration is in the study of particle physics. By using this group, physicists can analyze the behavior of particles with constant acceleration in the context of special relativity. Additionally, this group has applications in the field of general relativity, as it can be used to describe the behavior of accelerated objects in curved spacetime.

## 5. How does the maximal invariance group for constant acceleration differ from other symmetry groups?

The maximal invariance group for constant acceleration is unique compared to other symmetry groups. While other groups, such as the rotational symmetry group, describe the behavior of systems in specific scenarios, the maximal invariance group for constant acceleration is more general and can be applied to a wide range of systems with constant acceleration. This group also has a special connection to spacetime and special relativity, setting it apart from other symmetry groups.

Replies
18
Views
2K
Replies
25
Views
2K
Replies
36
Views
4K
Replies
144
Views
7K
Replies
22
Views
2K
Replies
6
Views
1K
Replies
10
Views
1K
Replies
13
Views
2K
Replies
7
Views
2K
Replies
75
Views
4K