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Difference between Riemannianpseudo-Riemannian metric and the solution of Einstein equationfield equations

NakaharNakahara in his book on the Geometry and Topology introduces the Riemannianpseudo-Riemannian metric as a type of (0,2) tensor which contains some properties which I interpreted them as a kind of multiplication factor. On the other hand, as we know, the metric is a solution of Einstein equationfield equations, hence it describes the gravitational field (maybe it is better to say that it is an auxiliary field since the connection could have a better description of this field on the equivalence principle.). I want to know, how can be connect these two definitions on metric? Why the factor that change the upper indices to lower one and vice versa, is known as a gravitational field?

Difference between Riemannian metric and the solution of Einstein equation

Nakahar in his book on the Geometry and Topology introduces the Riemannian metric as a type of (0,2) tensor which contains some properties which I interpreted them as a kind of multiplication factor. On the other hand, as we know, the metric is a solution of Einstein equation, hence it describes the gravitational field (maybe it is better to say that it is an auxiliary field since the connection could have a better description of this field on the equivalence principle.). I want to know, how can be connect these two definitions on metric? Why the factor that change the upper indices to lower one and vice versa, is known as a gravitational field?

Difference between pseudo-Riemannian metric and the solution of Einstein field equations

Nakahara in his book on the Geometry and Topology introduces the pseudo-Riemannian metric as a type of (0,2) tensor which contains some properties which I interpreted them as a kind of multiplication factor. On the other hand, as we know, the metric is a solution of Einstein field equations, hence it describes the gravitational field (maybe it is better to say that it is an auxiliary field since the connection could have a better description of this field on the equivalence principle.). I want to know, how can be connect these two definitions on metric? Why the factor that change the upper indices to lower one and vice versa, is known as a gravitational field?

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Nakahar in his book on the Geometry and Topology introduces the Riemannian metric as a type of (0,2) tensor which contains some properties which I interpreted them as a kind of multiplication factor. On the other hand, as we know, the metric is a solution of Einstein equation, hence it describes the gravitational field (maybe it is better to say that it is an auxiliary field since the connection could have a better description of this field on the equivalence principle.). I want to know, how can be connect these two definitions on metric? Why the factor that change the upper indices to lower one and vice versa, is known as a gravitational field?

Nakahar in his book on the Geometry and Topology introduces the Riemannian metric as a type of (0,2) tensor which contains some properties which I interpreted them as a kind of multiplication factor. On the other hand, as we know, the metric is a solution of Einstein equation, hence it describes the gravitational field (maybe it is better to say that it is an auxiliary field since the connection could have a better description of this field on the equivalence principle.). I want to know, how can be connect these two definitions on metric?

Nakahar in his book on the Geometry and Topology introduces the Riemannian metric as a type of (0,2) tensor which contains some properties which I interpreted them as a kind of multiplication factor. On the other hand, as we know, the metric is a solution of Einstein equation, hence it describes the gravitational field (maybe it is better to say that it is an auxiliary field since the connection could have a better description of this field on the equivalence principle.). I want to know, how can be connect these two definitions on metric? Why the factor that change the upper indices to lower one and vice versa, is known as a gravitational field?

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source | link

Difference between Riemannian metric and the solution of Einstein equation

Nakahar in his book on the Geometry and Topology introduces the Riemannian metric as a type of (0,2) tensor which contains some properties which I interpreted them as a kind of multiplication factor. On the other hand, as we know, the metric is a solution of Einstein equation, hence it describes the gravitational field (maybe it is better to say that it is an auxiliary field since the connection could have a better description of this field on the equivalence principle.). I want to know, how can be connect these two definitions on metric?