Gravitons and the Field Theory Metric

[QuantumClue]

We know the background metric has the description;

$$g_{\mu \nu}\sim\eta_{\mu \nu}+h_{\mu \nu}$$

I would like to know what the physical meaning is of the difference then?

$$h_{\mu \nu} - \eta_{\mu \nu}$$

When I've read field theories describing gravitons, they are usually denoted as:

$$h_{\mu \nu}$$

Where it can be seen as a perturbation on the background relation making it a fluctuation. I also know that the background metric can also be made to vanish by taking some form of diffeomorphism perhaps (?) not sure, I'll need to check that. But what is the significant meaning of taking the difference, does it produce a constant?

 

[AndyW]

The difference between h_{\mu \nu} and \eta_{\mu \nu} in the background metric has significant physical meaning in the context of general relativity. In this case, the background metric is taken to be the Minkowski metric, which is the flat, unchanging spacetime in special relativity. The Minkowski metric is represented by \eta_{\mu \nu} and is considered to be the "natural" or "unperturbed" state of spacetime.

The quantity h_{\mu \nu} represents the perturbation or deviation from this flat spacetime. In general relativity, gravitons are described as these perturbations in the spacetime metric. They can be thought of as ripples or fluctuations in the fabric of spacetime caused by the presence of massive objects.

Taking the difference between h_{\mu \nu} and \eta_{\mu \nu} allows us to isolate and study these perturbations separately from the background metric. This is important in understanding the effects of gravity and how it affects the motion of objects in spacetime.

Moreover, the difference between h_{\mu \nu} and \eta_{\mu \nu} is not a constant, but rather a function that varies with position and time. This function describes the strength and direction of the gravitational field at a given point in spacetime.

Finally, the ability to make the background metric vanish by using a diffeomorphism (a mathematical transformation) is a consequence of the principle of general covariance in general relativity. It allows us to choose a different reference frame or coordinate system to describe the same physical situation, without changing the underlying physical reality.

In summary, the difference between h_{\mu \nu} and \eta_{\mu \nu} in the background metric is crucial in understanding the effects of gravity and the behavior of spacetime in general relativity. It represents the perturbation or deviation from the flat, unchanging Minkowski metric and allows us to study the effects of gravity separately from the background.