Contracting Riemann tensor with itself

In summary, Padmanabhan discusses the horizon singularity of the Schwarzschild metric in chapter 8 of his book "Gravitation: Foundations and Frontiers." He argues that this singularity is only a coordinate singularity and attempts to show this by constructing a scalar from the Riemann tensor. However, the resulting equation (eq. 8.3) is incorrect and does not follow from the symmetries of the Riemann tensor. This scalar is known as the Kretschmann scalar and has been a topic of recent discussion.
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ShayanJ
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In chapter 8 of Padmanabhan's "Gravitation: Foundations and Frontiers" titiled Black Holes, where he wants to explain that the horizon singularity of the Schwarzschild metric is only a coordinate singularity, he does this by trying to find a scalar built from Riemann tensor and show that its well-behaved at the horizon. He writes (eq. 8.3) ##R_{abcd} R^{abcd}=\frac{48 M^2}{r^6}##. But this can't be correct. The (anti-)symmetries of the Riemann dictate that ##R_{abcd} R^{abcd}## is equal to zero. What is he doing here?
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The anti-symmetries do not dictate that this is zero.
 
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Also, this quantitiy is called the Kretschmann scalar. You may have seen posts about it recently.
 
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1. What is the Riemann tensor?

The Riemann tensor is a mathematical object used in differential geometry to describe the curvature of a space. It is defined as a tensor that encodes information about the intrinsic curvature of a manifold.

2. How do you contract the Riemann tensor with itself?

The Riemann tensor can be contracted with itself by performing a summation over two indices, resulting in a new tensor known as the Ricci tensor. This can then be further contracted to obtain the Ricci scalar, which is a scalar quantity used to measure the curvature of a space.

3. What is the physical significance of contracting the Riemann tensor with itself?

Contracting the Riemann tensor is a mathematical operation that yields important physical quantities used in general relativity, such as the Ricci tensor and scalar. These quantities help describe the curvature of space and are used to formulate Einstein's field equations, which govern the behavior of matter and energy in the presence of gravitation.

4. What are some applications of contracting the Riemann tensor with itself?

Contracting the Riemann tensor is a fundamental operation used in general relativity to study the curvature of space and the effects of gravity on matter and energy. It is also used in other fields, such as mathematical physics, string theory, and cosmology, to describe the behavior of spacetime on both small and large scales.

5. Are there any limitations to contracting the Riemann tensor with itself?

While contracting the Riemann tensor is a useful mathematical operation, it is limited in its ability to fully describe the curvature of a space. It only yields a limited set of physical quantities and does not capture the full complexity of the curvature tensor. Additionally, the Riemann tensor is only defined for spaces with a certain number of dimensions, limiting its applicability to higher-dimensional theories.

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