Line element of Schwarzschild

The notation SO(3) is used because the group consists of "special" orthogonal matrices, meaning that they have determinant 1 and thus do not reflect space. This is in contrast to the general orthogonal group O(3), which includes reflections.
  • #1
off-diagonal
30
0
I've read Schwarzschild paper and I don't understand his conditions

"The solution is spatially symmetric with respect to the origin of the co-ordinate system in the sense that one finds again the same solution when x,y,z are subjected to an orthogonal transformation(rotation)"


Could anyone explain me about this??


Thank you
 
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  • #2
He's assuming spherical or rotational symmetry. In other words, the solution/metric does not depend on which direction you're looking in (from the origin). This is a reasonable assumption for the metric around a star, since stars are to a high degree "round" =)
 
  • #3
off-diagonal said:
"The solution is spatially symmetric with respect to the origin of the co-ordinate system in the sense that one finds again the same solution when x,y,z are subjected to an orthogonal transformation(rotation)"

Hi off-diagonal! :smile:

"orthogonal" means that it's a member of SO(3), the three-dimensional group of speical orthogonal transformations.

Basically, SO(3) means all rotations.

See http://en.wikipedia.org/wiki/Rotation_group for more details:
The rotation group is often denoted SO(3) for reasons explained below.
 

1. What is the line element of Schwarzschild spacetime?

The line element of Schwarzschild spacetime is a mathematical expression that describes the geometry of spacetime around a non-rotating, spherically symmetric mass, such as a black hole. It is given by the equation:

ds² = -(1-2GM/r)dt² + (1-2GM/r)^-1dr² + r²(dθ² + sin²θdφ²)

2. What do the terms in the line element represent?

The first term (-dt²) represents the time component, the second term (dr²) represents the radial distance, and the third term (dθ² + sin²θdφ²) represents the angular components of spacetime. The coefficient (1-2GM/r) is known as the Schwarzschild metric, which determines the curvature of spacetime around the massive object.

3. How does the line element relate to the concept of gravitational time dilation?

The line element contains a term (-dt²) that represents the time component of spacetime. This term is multiplied by the Schwarzschild metric, which increases as the distance from the massive object decreases. This means that time runs slower closer to the massive object, resulting in gravitational time dilation.

4. Can the line element be used to calculate the event horizon of a black hole?

Yes, the event horizon of a black hole is determined by the radius at which the Schwarzschild metric becomes equal to 0. This radius is known as the Schwarzschild radius and can be calculated using the line element. Any object within this radius will be unable to escape the gravitational pull of the black hole.

5. How does the line element of Schwarzschild differ from the line element of Minkowski spacetime?

The line element of Minkowski spacetime is given by ds² = -c²dt² + dx² + dy² + dz², where c is the speed of light. This line element describes a flat, non-curved spacetime. On the other hand, the line element of Schwarzschild spacetime contains additional terms that account for the curvature of spacetime caused by a massive object. Additionally, the Schwarzschild line element includes a time component that is affected by the mass of the object, while the Minkowski line element assumes a constant speed of light for all observers.

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