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unscientific
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Contracting \mu & \alpha - What Does It Mean?
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In summary, contraction of a tensor or product of tensors with an upper and a lower index free results in a sum of terms that depends on the indices involved. The second step, multiplying by ##g^{\nu \gamma}##, is also a contraction.
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PeterDonis
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Contraction is an operation that can be applied to any tensor or product of tensors with an upper and a lower index free. (In this case the upper index is ##\mu## and the lower index is ##\alpha##.) The contraction is just a sum over all tensor components for which ##\mu## and ##\alpha## take the same value. So, for example, a tensor ##T^{\mu}{}_{\alpha}## with one upper and one lower index can be contracted to a scalar ##T = T^0{}_0 + T^1{}_1 + T^2{}_2 + T^3{}_3##.
The contraction of the Bianchi identity has more terms because the identity itself has three terms, and each one becomes a sum of four terms when contracted. Note that the second step, multiplying by ##g^{\nu \gamma}##, is also a contraction, because the indexes ##\nu## and ##\gamma## appear as lower indexes in the Bianchi identity.
The contraction of the Bianchi identity has more terms because the identity itself has three terms, and each one becomes a sum of four terms when contracted. Note that the second step, multiplying by ##g^{\nu \gamma}##, is also a contraction, because the indexes ##\nu## and ##\gamma## appear as lower indexes in the Bianchi identity.
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unscientific
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PeterDonis said:
Do you mind showing the steps leading to the final result ## 2\nabla_v R_\beta^v -\nabla_\beta R = 0##?
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PeterDonis
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We start with
$$
\nabla_{\gamma} R^{\mu}{}_{\nu \alpha \beta} + \nabla_{\beta} R^{\mu}{}_{\nu \gamma \alpha} + \nabla_{\alpha} R^{\mu}{}_{\nu \beta \gamma} = 0
$$
Contracting ##\mu## and ##\alpha## gives (note that an index that is repeated, once upper and once lower, is summed over as I described; this is called the "Einstein summation convention" and is extremely useful):
$$
\nabla_{\gamma} R^{\alpha}{}_{\nu \alpha \beta} + \nabla_{\beta} R^{\alpha}{}_{\nu \gamma \alpha} + \nabla_{\alpha} R^{\alpha}{}_{\nu \beta \gamma} = 0
$$
Completing the contractions (meaning, collapsing the contracted indexes and using the fact that contracting the Riemann tensor on the upper and the middle lower index gives the Ricci tensor) gives:
$$
\nabla_{\gamma} R_{\nu \beta} - \nabla_{\beta} R_{\nu \gamma} + \nabla_{\alpha} R^{\alpha}{}_{\nu \beta \gamma} = 0
$$
where the minus sign in the second term comes in because the ##\alpha## index was the last lower index, not the middle one; swapping the indexes flips the sign (because the Riemann tensor is antisymmetric in the last two lower indexes). Now we contract with ##g^{\nu \gamma}## to give:
$$
g^{\nu \gamma} \nabla_{\gamma} R_{\nu \beta} - g^{\nu \gamma} \nabla_{\beta} R_{\nu \gamma} + g^{\nu \gamma} \nabla_{\alpha} R^{\alpha}{}_{\nu \beta \gamma} = 0
$$
Completing the contractions gives (note that ##R = g^{\nu \gamma} R_{\nu \gamma}## is the Ricci scalar):
$$
\nabla^{\nu} R_{\nu \beta} - \nabla_{\beta} R + \nabla_{\alpha} R^{\alpha}{}_{\beta} = 0
$$
The first and third terms are really the same thing, because the contracted index is a "dummy" index and we can relabel it freely, and we can also freely "flip" the indexes in the contraction (to put the upper index on ##R## and the lower index on ##\nabla## in the first term). This gives what we were looking for:
$$
2 \nabla_{\nu} R^{\nu}{}_{\beta} - \nabla_{\beta} R = 0
$$
$$
\nabla_{\gamma} R^{\mu}{}_{\nu \alpha \beta} + \nabla_{\beta} R^{\mu}{}_{\nu \gamma \alpha} + \nabla_{\alpha} R^{\mu}{}_{\nu \beta \gamma} = 0
$$
Contracting ##\mu## and ##\alpha## gives (note that an index that is repeated, once upper and once lower, is summed over as I described; this is called the "Einstein summation convention" and is extremely useful):
$$
\nabla_{\gamma} R^{\alpha}{}_{\nu \alpha \beta} + \nabla_{\beta} R^{\alpha}{}_{\nu \gamma \alpha} + \nabla_{\alpha} R^{\alpha}{}_{\nu \beta \gamma} = 0
$$
Completing the contractions (meaning, collapsing the contracted indexes and using the fact that contracting the Riemann tensor on the upper and the middle lower index gives the Ricci tensor) gives:
$$
\nabla_{\gamma} R_{\nu \beta} - \nabla_{\beta} R_{\nu \gamma} + \nabla_{\alpha} R^{\alpha}{}_{\nu \beta \gamma} = 0
$$
where the minus sign in the second term comes in because the ##\alpha## index was the last lower index, not the middle one; swapping the indexes flips the sign (because the Riemann tensor is antisymmetric in the last two lower indexes). Now we contract with ##g^{\nu \gamma}## to give:
$$
g^{\nu \gamma} \nabla_{\gamma} R_{\nu \beta} - g^{\nu \gamma} \nabla_{\beta} R_{\nu \gamma} + g^{\nu \gamma} \nabla_{\alpha} R^{\alpha}{}_{\nu \beta \gamma} = 0
$$
Completing the contractions gives (note that ##R = g^{\nu \gamma} R_{\nu \gamma}## is the Ricci scalar):
$$
\nabla^{\nu} R_{\nu \beta} - \nabla_{\beta} R + \nabla_{\alpha} R^{\alpha}{}_{\beta} = 0
$$
The first and third terms are really the same thing, because the contracted index is a "dummy" index and we can relabel it freely, and we can also freely "flip" the indexes in the contraction (to put the upper index on ##R## and the lower index on ##\nabla## in the first term). This gives what we were looking for:
$$
2 \nabla_{\nu} R^{\nu}{}_{\beta} - \nabla_{\beta} R = 0
$$
FAQ: Contracting \mu & \alpha - What Does It Mean?
What is contracting &mu &alpha and what does it mean?
Contracting &mu &alpha is a scientific term used to describe the process of reducing the size or quantity of something. In physics, it refers to the process of contracting or shrinking the dimensions of a physical object or system. In statistics, it refers to the process of reducing the variability or randomness in a data set.
How is contracting &mu &alpha different from expanding &mu &alpha?
While contracting &mu &alpha refers to the reduction or shrinking of something, expanding &mu &alpha refers to the opposite process of increasing the size or quantity of something. In physics, expanding &mu &alpha is often associated with the expansion of the universe, while in statistics it can refer to the increase in variability or randomness in a data set.
What are some examples of contracting &mu &alpha in science?
One example of contracting &mu &alpha in science is the process of cooling a gas, which causes it to contract and decrease in volume. Another example is the contraction of a muscle during exercise, which results in a decrease in its size. In statistics, contracting &mu &alpha can be seen in the process of data standardization, where the variability of data is reduced by transforming it to a common scale.
What are the applications of contracting &mu &alpha in research?
Contracting &mu &alpha has various applications in different fields of research. In physics, it is used in studies of the expansion of the universe, while in biology it is used to understand the contraction of muscles and the shrinking of cells. In statistics, contracting &mu &alpha is used for data analysis and data reduction, allowing researchers to identify patterns and relationships in a data set.
How does contracting &mu &alpha impact the interpretation of data?
Contracting &mu &alpha can have a significant impact on the interpretation of data. In physics, it can provide insights into the structure and evolution of the universe. In biology, it can help understand the functioning of muscles and cells. In statistics, contracting &mu &alpha can help identify significant trends and relationships in a data set by reducing the effect of random variability. This can lead to more accurate and reliable conclusions in research.