# Time-Dependent Lagrangian Leads to Time Dilation?

• I
• stevendaryl
In summary, the relativistic Lagrangian for a particle moving under a scalar potential leads to equations of motion that can be seen as a generalization of Newton's F=ma. However, a difference is that it is a 4-D equation and the 0th component shows that time dilation can depend not only on velocity but also on the potential. This time dilation can be positive or negative, leading to time running faster or slower for the particle. While this effect may be difficult to test due to the limited availability of scalar fields, it has been studied in more detail and published in a book chapter. The scalar potential can also be viewed as a dynamical mass squared, allowing for particles to exceed the speed of light. However,
stevendaryl
Staff Emeritus
This is just something unexpected that I noticed recently, and I hadn't heard anyone mention it before.

The relativistic Lagrangian for a particle moving under a scalar potential ##\Phi## is this:

##L = \frac{1}{2} m g_{\mu \nu} \dfrac{dx^\mu}{d\tau} \dfrac{dx^\nu}{d\tau} - \Phi##

This leads to the equations of motion:

##m \dfrac{d^2 x^\mu}{d\tau^2} = - \partial^\mu \Phi##

So that's just the relativist generalization of Newton's ##F = m A##, with ##F = -\nabla \Phi##. However, a difference is that it's a 4-D equation, rather than a 3-D equation. So let's look at just the 0th component, with ##x^0 = t##:

##m \dfrac{d^2 t}{d\tau^2} = - \dfrac{\partial \Phi}{\partial t}##

This is truly unexpected (to me). When there is no potential, ##\dfrac{dt}{d\tau}## is the time dilation factor ##\gamma##. The above equation seems to be saying that time dilation depends not only on velocity (or spacetime curvature, if you consider General Relativity, which I'm not doing here) but also on the potential. So even a particle at rest will experience time dilation if it is in a time-varying potential.

Another thing that is surprising is that this time dilation can be positive or negative. So if a particle starts out at rest, with ##\dfrac{dt}{d\tau} = 1##, then a negative value for ##- \dfrac{\partial \Phi}{\partial t}## will lead to the particle having ##\dfrac{dt}{d\tau} \gt 1##. So time runs faster for the particle, rather than slower.

Is this a real effect? My guess is that it wouldn't be easy to test because there are so few scalar fields (the only one I know of is the Higgs field), and they are not as easily manipulated as the electromagnetic field.

Demystifier and Dale
I have studied this stuff in more detail in
https://arxiv.org/abs/1006.1986
and published as a part of a book chapter
https://arxiv.org/abs/1205.1992

The scalar potential can be viewed as a dynamical mass squared, which can become negative so that particle can exceed the velocity of light.

But note that your parameter ##\tau##, called ##s## in my work, is not the usual proper time. Hence, the potential does not modify the time dilation. It only modifies the relation between proper time and this parameter.

## 1. What is time-dependent Lagrangian?

Time-dependent Lagrangian is a mathematical framework used in physics to describe the motion of a system over time. It takes into account the kinetic and potential energies of the system and can be used to derive equations of motion.

## 2. How does time-dependent Lagrangian lead to time dilation?

According to the theory of relativity, time dilation occurs when an object is moving at high speeds relative to an observer. Time-dependent Lagrangian takes into account the speed of an object in its equations, and as the object's speed increases, time dilation occurs.

## 3. Can you provide an example of time dilation caused by time-dependent Lagrangian?

One example of time dilation caused by time-dependent Lagrangian is the famous "twin paradox." In this scenario, one twin stays on Earth while the other travels in a spaceship at high speeds. The twin in the spaceship will experience time dilation due to their high speed, causing them to age slower than the twin on Earth.

## 4. How is time-dependent Lagrangian related to the theory of relativity?

Time-dependent Lagrangian is a mathematical tool used in the theory of relativity to describe the effects of time dilation. It is based on the principle of least action, which is a fundamental concept in the theory of relativity.

## 5. Are there any practical applications of time-dependent Lagrangian and time dilation?

Yes, time-dependent Lagrangian and time dilation have several practical applications in modern technology. For example, they are used in GPS systems to account for the effects of time dilation caused by the high speeds of satellites. They are also used in particle accelerators and space travel to accurately predict and account for the effects of time dilation.

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