Riemann tensor and flat spacetime

Thread starter Ranku
Start date Apr 19, 2010
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Flat Riemann Riemann tensor Spacetime Tensor

In summary, the Riemann tensor is a 4th order tensor that describes the curvature of spacetime in the theory of general relativity. It is related to flat spacetime, where it is zero, and is important in general relativity as it helps calculate the gravitational field and predict the motion of objects. The Riemann tensor is calculated using the Christoffel symbols, but cannot be directly measured or observed as it is a mathematical concept. However, its effects can be observed through the motion of objects in the presence of gravity.

Apr 19, 2010

Ranku

When Riemann tensor = 0, spacetime is flat. Is the geometry of this flat spacetime that of special relativity?

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Apr 19, 2010

haushofer

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That depends on your theory. In normal GR, it is.

Apr 19, 2010

Ranku

haushofer said:

That depends on your theory. In normal GR, it is.

Ok. Thanks.

1. What is the Riemann tensor?

The Riemann tensor, also known as the Riemann curvature tensor, is a mathematical object that describes the curvature of spacetime in the theory of general relativity. It is a 4th order tensor that contains information about the curvature of spacetime in all directions at a given point.

2. How is the Riemann tensor related to flat spacetime?

In flat spacetime, the Riemann tensor is zero. This means that the curvature of spacetime is also zero, and the laws of physics are the same in all directions. In other words, flat spacetime is a special case where the Riemann tensor vanishes, and the spacetime is considered to be "flat" or "unbent".

3. Why is the Riemann tensor important in general relativity?

The Riemann tensor is an essential tool in the theory of general relativity because it describes the curvature of spacetime, which is directly related to the presence of matter and energy. This tensor is used to calculate the gravitational field and predict how objects will move in curved spacetime.

4. How is the Riemann tensor calculated?

The Riemann tensor is calculated using the Christoffel symbols, which are derived from the metric tensor. The metric tensor contains information about the geometry of spacetime, and the Christoffel symbols are mathematical functions that describe how the metric changes from one point to another. By combining these two objects, the Riemann tensor can be calculated.

5. Can the Riemann tensor be measured or observed?

No, the Riemann tensor cannot be directly measured or observed. It is a mathematical concept used in the theory of general relativity to describe the curvature of spacetime. However, its effects can be observed through the motion of objects in the presence of gravity, such as the orbit of planets around a star, which is a result of the curvature of spacetime caused by the mass of the star.