Kronecker Delta: A Relativity and Tensor Explanation

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SUMMARY

The Kronecker delta, denoted as \(\delta_{ij}\), is a mathematical function that returns 1 when two integers are equal and 0 otherwise. This function is crucial in tensor analysis, particularly in the context of relativity, as it simplifies expressions involving summation over repeated indices. For example, in the equation \(x^\mu \delta_{\mu \nu} = x^\nu\), the Kronecker delta effectively filters terms, allowing for concise representation of tensor components. Its role as a metric tensor in Euclidean space further underscores its significance in mathematical physics.

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  • Understanding of tensor analysis
  • Familiarity with the summation convention
  • Basic knowledge of relativity concepts
  • Mathematical notation for functions and matrices
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This discussion is beneficial for physicists, mathematicians, and students studying relativity and tensor analysis, particularly those looking to deepen their understanding of mathematical functions used in these fields.

Terilien
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I keep seeing this come up in relativity and tensor resources but I have no idea wht the heck it means. Could someone explain it to me?
 
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Terilien said:
I keep seeing this come up in relativity and tensor resources but I have no idea wht the heck it means. Could someone explain it to me?

The Kronecker delta is a function of two integers. If the integers are the same then the value of the function is 1. Otherwise it is zero. This function can be represented as a matrix. The notation for this function is \deltaij.

Pete
 
pmb_phy said:
The Kronecker delta is a function of two integers. If the integers are the same then the value of the function is 1. Otherwise it is zero. This function can be represented as a matrix. The notation for this function is \deltaij.

Pete


Why is it important in tensor analysis?
 
Example from relativity. Let the coordinates of an event be \left\{x^0 , x^1 , x^2 , x^3 \right\}. Then, using the summation convention of summing over repeated indices,

x^\mu \delta_{\mu \nu} = x^0 \delta_{0 \nu} + x^1 \delta_{1 \nu} + x^2 \delta_{2 \nu} + x^3 \delta_{3 \nu}.

Since the Kronecker delta is zero unless both indices are equal, only one of the terms in the above sum survives. We don't know which one, but we know it's the one that has \nu as its first index. Therefore, the sum equals x^\nu .
 
Last edited:
Terilien said:
Why is it important in tensor analysis?
because it's a metric tensor of euclidean space? dunno. the "importance" asigned to things by different people is quite biased.
 

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