What is the physical meaning of curvature?

In summary, the conversation discusses the relationship between curvature and potential energy, specifically in the case of a parabolic potential. The mathematical definition of curvature is provided and it is noted that the resulting equation resembles a lorenzian line shape, which is the distribution function of a harmonic oscillator in a parabolic potential. The question is posed whether this resemblance is a coincidence or if there is a physical relationship between the two. The response suggests that there is a relationship, but it is difficult to define and understand. The concept of parallel transport is mentioned as a potential explanation for this relationship.
  • #1
cnelson
5
0
I'm not sure if this belongs here or in the physics section. The mathematical definition of curvature is the derivative of the unit tangent vector normalized to the arc length: [itex]\kappa[/itex] = [itex]\frac{dT}{ds}[/itex]. If we apply this to a parabola with equation y = [itex]x^{2}[/itex] we get [itex]\frac{2}{(1+4x^{2})^{3/2}}[/itex]. This resembles a lorenzian line shape which is the distribution function (amplitude vs frequency) of a harmonic oscillator in a parabolic potential (i.e. a graph of amplitude vs frequency shows a resonance at some frequency.)

My question:

Is the curvature of a potential energy function related to the distribution function of the potential energy function or does it have some other physical relationship that I am missing? Or is the resemblance of the curvature of a parabola and the lorenzian line shape a coincidence?

Sorry if this is kind of vague let me know if you have questions.
 
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  • #2
The curvature of potential energy is definitely related to distribution function, but the dependence is weird.I can't give any physical meaning to curvature which might suit the dependence.
 
  • #3
The curvature of a surface is (or at least can be) defined as moving a vector through parallel transport around a closed loop.
 
  • #4
Thanks for the responses. I'm not familiar with parallel transport so I'll study up and see if it can answer my question.
 

1. What is curvature in physics?

Curvature in physics refers to the amount of bending or distortion present in a physical system. It can also be thought of as the measure of how much a curve deviates from being a straight line. In the context of general relativity, curvature is used to describe the warping of space-time caused by the presence of massive objects.

2. What is the physical meaning of curvature?

The physical meaning of curvature is that it represents the influence of gravity on the shape of space-time. In a region with high curvature, objects will experience a greater gravitational pull towards the center of the curvature. This is why massive objects, such as planets and stars, have a significant impact on the curvature of space-time.

3. How is curvature measured?

Curvature is measured using mathematical tools such as tensors and differential geometry. In general relativity, the curvature of space-time is described by the Riemann curvature tensor, which contains information about the curvature at every point in space-time.

4. What does a positive or negative curvature mean?

In the context of general relativity, a positive curvature means that space-time is curved in a way that causes objects to be pulled towards the center of the curvature. This is the case for a massive object like a planet. On the other hand, a negative curvature means that space-time is curved in a way that causes objects to be pushed away from the center of the curvature, as is the case for dark energy.

5. How does curvature affect the motion of objects?

Curvature affects the motion of objects by influencing the path that they take through space-time. In a region with high curvature, the path of an object will be more curved, leading to a change in its trajectory. This is why objects in orbit around a massive object like a planet follow a curved path instead of a straight line.

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