# Calculate the invariant problem

• Nusc
In summary, the task is to calculate the invariant E^{\alpha \beta} E_{\alpha \beta} by applying the metric and using the equation g_{\alpha n} g_{\beta m} E_{n m} E^{n m}. However, this equation is incorrect as it includes both dummy and free indices and repeats the indices m and n. The correct equation should only involve either E^{\alpha \beta} or E_{\alpha \beta}.
Nusc

## Homework Statement

Calculate the invariant
$$E^{\alpha \beta} E_{\alpha \beta}$$

## The Attempt at a Solution

we apply the metric in this case,
$$E^{\alpha \beta} E_{\alpha \beta} = g_{\alpha n} g_{\beta m} E_{n m} E^{n m}$$

is that even correct?

Nusc said:
we apply the metric in this case,
$$E^{\alpha \beta} E_{\alpha \beta} = g_{\alpha n} g_{\beta m} E_{n m} E^{n m}$$

is that even correct?

No, that equation is non-sensical...on the lefthandside $\alpha$ and $\beta$ are dummy indices (they are being summed over)...on the righthandside they are free indices and you also have $m$ and $n$ appearing 3 times (which is notational nonsense).

I assume you are given either $E^{\alpha \beta}$ or $E_{\alpha \beta}$?

## 1. What is the "invariant problem" and why is it important?

The invariant problem is a mathematical concept that involves identifying and understanding properties or quantities that do not change under certain transformations or operations. It is important because it helps us to simplify complex problems and identify key relationships between variables.

## 2. How do you calculate the invariant problem?

The exact method for calculating the invariant problem will depend on the specific problem at hand. However, in general, it involves identifying the variables and relationships involved, and then using mathematical operations to simplify the problem and identify the invariant properties.

## 3. Can you give an example of the invariant problem?

One example of the invariant problem is the conservation of energy in physics. In this case, the total amount of energy in a closed system remains constant, regardless of any transformations or changes that may occur within the system.

## 4. How is the invariant problem used in scientific research?

The invariant problem is used in scientific research to identify and understand key relationships and properties in complex systems. It can help scientists to simplify and solve problems, as well as make predictions and test hypotheses.

## 5. Are there any limitations to using the invariant problem in scientific research?

While the invariant problem can be a useful tool in scientific research, it does have some limitations. In some cases, it may not fully capture all aspects of a complex system, and there may be exceptions or special cases that do not follow the invariant properties. Additionally, the calculations involved can be quite complex and time-consuming.

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