Finding Constant \alpha_M in SET Definition

In summary, Wald defines the stress-energy for an arbitrary matter field as T_{ab}=-\frac{\alpha_M}{8\pi} \frac{1}{ \sqrt{-g}} \frac{\delta S_M}{\delta g^{ab}}, where S_M is the action for the particular type of matter field and \alpha_M is a constant that determines the form of the Lagrangian in the coupled Einstein-matter field equations. The value of \alpha_M can be found using different conventions, such as taking \alpha_{KG}=16\pi for a Klein-Gordon field and \alpha_{EM}=4 for an EM field. However, there is no specific prescription for finding the value of \alpha_M for an
  • #1
elfmotat
260
2
So I was looking through Wald when I noticed his definition of the stress-energy for an arbitrary matter field:

[tex]T_{ab}=-\frac{\alpha_M}{8\pi} \frac{1}{ \sqrt{-g}} \frac{\delta S_M}{\delta g^{ab}}[/tex]

where [itex]S_M[/itex] is the action for the particular type of matter field being considered, and [itex]\alpha_M[/itex] is some constant that determines the form of the Lagrangian for the coupled Einstein-matter field equations:

[tex]\mathcal{L}=R\sqrt{-g}+\alpha_M \mathcal{L}_M[/tex]

For example, for a Klein-Gordon field we take [itex]\alpha_{KG}=16\pi[/itex], and for an EM field we take [itex]\alpha_{EM}=4[/itex]. Now, my question is whether or not there is some prescription for finding the value of [itex]\alpha_M[/itex]. How could I go about finding [itex]\alpha_M[/itex] for an arbitrary [itex]\mathcal{L}_M[/itex]?

I feel like I'm missing something painfully obvious.
 
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  • #2
I guess everybody has his own conventions. The usual ones follow.

The Einstein Equations are Gμν = 8πG Tμν. To get this equation we use an action I = IG + IM where IG = (1/16πG) ∫√-g R d4x and Tμν = (2/√-g) δIM/δgμν.

For electromagnetism, L = (-1/4)FμνFμν. This is in Heaviside units where e2/4πħc = 1/137. In Gaussian units where e2/ħc = 1/137, the Lagrangian would instead be IM = (-1/16π)FμνFμν.
 

FAQ: Finding Constant \alpha_M in SET Definition

1. What is the purpose of finding the constant αM in SET definition?

The constant αM in SET definition is used to determine the probability of a single event occurring in a set. It is a measure of uncertainty or randomness, and is essential in various fields such as statistics, probability, and decision making.

2. How is the constant αM calculated in SET definition?

The constant αM is calculated by dividing the number of outcomes in a set that satisfy a certain condition by the total number of possible outcomes in that set. This can be represented mathematically as αM = # of favorable outcomes / # of total outcomes.

3. What is the range of values for the constant αM in SET definition?

The constant αM can have a range of values from 0 to 1, where 0 represents impossibility and 1 represents certainty. A value closer to 0 indicates a lower probability of the event occurring, while a value closer to 1 indicates a higher probability.

4. How is the constant αM used in decision making?

The constant αM is used in decision making by providing a measure of the likelihood of a certain outcome. It can help in comparing different options and making informed decisions based on the probability of success or failure in a particular scenario.

5. Are there any limitations to finding the constant αM in SET definition?

Yes, there are some limitations to finding the constant αM in SET definition. It assumes that all outcomes in a set are equally likely, which may not always be the case in real-world situations. It also does not take into account external factors that may affect the probability of an event occurring.

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