Understanding the Relationship between Connection and Metric in Curved Spacetime

In summary, in a curved spacetime, a change of metric leads to a change of connection. This is because parallel transported vectors and geodesics are not preserved under changes of metric, hence the connection must also change. To convert a tensor from one connection to another, one must re-calculate the Christoffel symbols with the new metric.
  • #1
gnieddu
24
1
Hi,

I'm struggling to grasp the physical reason behind the fact that, in a curved spacetime, a change of metric implies, in general, a change of connection, i.e. if I have two metrics [tex]g_{ab}[/tex] and [tex]\hat{g}_{ab}[/tex], in general [tex]\nabla_a \neq \hat{\nabla}_a[/tex].

Besides this, is there any relationship between the two connections? In other words, if I know [tex]\nabla_aT[/tex] for a given tensor T, is there a general formula which converts it into [tex]\hat{\nabla}_aT[/tex]?

Thanks
 
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  • #2
There isn't such a thing as "the" connection. There are many possible connections. In general relativity, the connection is chosen to be the Levi-Civita connection, which is defined by the metric.
 
  • #3
gnieddu said:
Hi,

I'm struggling to grasp the physical reason behind the fact that, in a curved spacetime, a change of metric implies, in general, a change of connection, i.e. if I have two metrics [tex]g_{ab}[/tex] and [tex]\hat{g}_{ab}[/tex], in general [tex]\nabla_a \neq \hat{\nabla}_a[/tex].

Imagine stretching a surface embedded in Euclidean space in a non-uniform way. This is a change of the induced metric on the surface, and you can see that parallel transported vectors on the original surface are no longer parallel transported. This is why the connection changes when the metric is changed. Another way to see it is to imagine taking a flat surface with straight lines drawn on it, and then stretching it over a sphere. There are many ways to do this, and in generel, the straight lines need not become great circles on the sphere, i.e. they are no longer geodesics with the new induced metric. Since the set of geodesics determine the connection, and since geodesics are not preserved by changes of metric, the connection must change.

gnieddu said:
Besides this, is there any relationship between the two connections? In other words, if I know [tex]\nabla_aT[/tex] for a given tensor T, is there a general formula which converts it into [tex]\hat{\nabla}_aT[/tex]?

Thanks

You simply have to re-calculate the Christoffel symbols with the new metric.
 
Last edited:
  • #4
Thanks!
 

1. What is the concept of connection in curved spacetime?

The concept of connection in curved spacetime is a mathematical tool used to describe how objects move in a curved space. It takes into account the curvature and shape of the space, as well as the motion of objects within that space. In other words, it describes how objects are connected to each other in a curved space.

2. How is the connection related to the metric in curved spacetime?

The connection and the metric in curved spacetime are closely related. The metric defines the distance between two points in a curved space, while the connection describes how objects move in that space. The connection is derived from the metric, and it is used to calculate the curvature of the space.

3. What is the role of the connection in general relativity?

The connection plays a critical role in general relativity as it is used to describe the curvature of spacetime. In Einstein's theory of general relativity, the curvature of spacetime is related to the distribution of matter and energy. The connection allows us to calculate the curvature and understand how objects move in this curved space.

4. How does the relationship between connection and metric affect the behavior of light in curved spacetime?

The relationship between connection and metric affects the behavior of light in curved spacetime through the phenomenon of gravitational lensing. The curvature of spacetime bends the path of light, and this bending is described by the connection and metric. This is why we see images of distant galaxies distorted when they pass through a massive object, such as a galaxy or cluster of galaxies.

5. Can the relationship between connection and metric be visualized in curved spacetime?

Yes, the relationship between connection and metric can be visualized in curved spacetime through the use of mathematical models and computer simulations. These models and simulations help us understand the effects of gravity and the behavior of objects in a curved space. However, it is important to note that these visualizations are based on mathematical equations and do not represent the actual physical space.

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