# Can equation of motion depend on the choice of metric?

• arroy_0205
In summary, the equation of motion for a scalar field depends on the choice of metric, which can lead to different signs in the kinetic term and potential term. To ensure a unique equation of motion, the signs of the kinetic term must be opposite in the two different metric conventions. This change in sign also applies to the Lagrangian, making it a symmetry of the system.

#### arroy_0205

It seems strange to me that in such a simple case as the following the equation of motion depends on the choice of metric! In flat space we have say:
$$\partial^{\mu}\partial_{\mu}\phi=\frac{dV}{d\phi}$$
Suppose \phi depends on space alone and we work in 1+1 dimension. then
$$\eta_{\mu \nu}=\left (-1,1 \right ) \rightarrow \frac{d^2\phi}{dx^2}=\frac{dV}{d\phi}$$
and
$$\eta_{\mu \nu}=\left ( +1,-1 \right ) \rightarrow -\frac{d^2\phi}{dx^2}=\frac{dV}{d\phi}$$
What is wrong here? Obviously the field configuration can not depend on the metric choice! Generally particle physicists and cosmologists use opposite signatures. Can anybody please point out?

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I want to add: It seems the only way to get unique equation of motion is to choose signs of kinetic term in the Lagrangian density opposite in these two cases (that is for two different metric conventions). Is that true?
$$L=\pm \frac{1}{2}\eta^{\mu \nu}\partial_{\mu}\phi \partial_{\nu}\phi -V(\phi)$$

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You have to change by an overall sign all the Lagrangian. In your example this turns out a change into the sign of V. Indeed, this is a symmetry of the Lagrangian as you can change all by an overall sign and you always get an extremum giving the right equation of motion (you have not to care about this being a minimum or a maximum). This means that when you choose a metric or the other a change into the sign of the potential is required.

Jon

arroy_0205 said:
I want to add: It seems the only way to get unique equation of motion is to choose signs of kinetic term in the Lagrangian density opposite in these two cases (that is for two different metric conventions). Is that true?
$$L=\pm \frac{1}{2}\eta^{\mu \nu}\partial_{\mu}\phi \partial_{\nu}\phi -V(\phi)$$

Yes, and this makes sense.

If $\eta = - \eta'$, and

$$L = \frac{1}{2}\eta^{\mu \nu}\partial_{\mu}\phi \partial_{\nu}\phi -V \left(\phi\right),$$

then

$$L = - \frac{1}{2}\eta'{}^{\mu \nu}\partial_{\mu}\phi \partial_{\nu}\phi -V \left(\phi\right).$$

The same change of sign occurs when the scalar proper time is expressed in terms of spacetime metrics that have different sign conventions.

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## 1. Can the equation of motion depend on the choice of metric?

Yes, the equation of motion can depend on the choice of metric. The metric is a mathematical representation of the underlying geometry of a space and it affects the way that objects move through that space. Therefore, different choices of metric can result in different equations of motion.

## 2. How does the choice of metric affect the equation of motion?

The choice of metric determines the geometry of the space in which the equation of motion is being considered. This, in turn, affects the curvature of the space and the behavior of objects moving through it, ultimately leading to different equations of motion.

## 3. Can different metrics lead to the same equation of motion?

Yes, it is possible for different metrics to result in the same equation of motion. This can occur when the metrics have similar underlying geometries, even though they may be described differently mathematically. Additionally, some metrics may be related by a transformation that preserves the equation of motion.

## 4. How do scientists choose which metric to use in studying the equation of motion?

Scientists typically choose a metric based on the specific properties of the space they are studying and the phenomenon they are trying to model. They may also consider mathematical simplicity and elegance in their choice of metric.

## 5. Can the equation of motion be solved for any choice of metric?

Yes, the equation of motion can be solved for any choice of metric. However, the complexity of the solution may vary depending on the metric chosen. Some metrics may result in simpler or more elegant solutions, while others may require more complex mathematical methods to solve.