# Four-Vector Physics: Exploring Questions

• actionintegral
In summary, the conversation discusses the concept of four-vectors and their components, including the four-potential, four-gradient, and four-momentum. The question is posed whether the negative four-gradient of the four-potential is equal to the time derivative of the four-momentum, but it is determined that the indices do not balance and there may be a conventional choice of sign involved. It is noted that the Lorentz Force expression may be relevant for understanding this relationship and that charge plays a significant role. Finally, the electromagnetic potential and its relationship to the four-momentum are explored in more detail.
actionintegral
I am trying to get a handle on four-vectors. I see that there is a thing called a four-potential, a four-gradient, and a four momentum.

Is it reasonable to ask if the negative four-gradient of the four-potential is equal to the time derivative of the four-momentum?

actionintegral said:
I am trying to get a handle on four-vectors. I see that there is a thing called a four-potential, a four-gradient, and a four momentum.

Is it reasonable to ask if the negative four-gradient of the four-potential is equal to the time derivative of the four-momentum?

No. Transcribing what you said: $$-\nabla_a A_b \stackrel{?}{=} \frac{d}{dt} p_a$$. The indices don't balance.

(Note the four-potential $$A_a$$ refers to the electromagnetic potential, which decomposes in an observer's coordinate system into the scalar potential $$\phi$$ and the vector potential $$\vec A$$.)

By "time derivative", do you mean derivative with respect to proper-time?

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The "four-potential" must be the electromagnetic potential $A_\mu$. Try writing down your second sentence as a tensor equation, it doesn't make sense, and the time derivitive of the momentum is not covariant, you need to use the proper time $\tau$. However, IIRC

$\frac{dp_\mu}{d\tau} = (\partial_\mu A_\nu - \partial_\nu A_\mu)u^\nu$

where $u^\nu$ is the 4-velocity of a body. But note what the space part of this equation reduces to in the rest frame of the body.

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Daverz said:
The "four-potential" must be the electromagnetic potential $A_\mu$. Try writing down your second sentence as a tensor equation, it doesn't make sense, and the time derivitive of the momentum is not covariant, you need to use the proper time $\tau$. However, IIRC

$\frac{dp_\mu}{d\tau} = (\partial_\mu A_\nu - \partial_\nu A_\mu)u^\nu$

where $u^\nu$ is the 4-velocity.

There's a $$q$$ on the right hand side and, possibly, a conventional choice of sign. This is the Lorentz Force expression on the right.

Note: $$\frac{dp_b}{d\tau}=u^a\nabla_a p_{b}=m u^a\nabla_a u_{b}$$.

Yeah, I sort of forgot that charge stuff. Kind of important if you want there to be any force on the body. Looking it up this time (Ohanian, Gravitation and Spacetime, 2nd ed, pp. 95-97:

$A^\mu = (\phi, A_x, A_y, A_z)$

$F^{\mu\nu} = \partial^\mu A^\nu - \partial^\nu A^\mu$

$\frac{dp^\mu}{d\tau} = \frac{q}{m}p_\nu F^{\mu \nu}$

## 1. What is a four-vector in physics?

A four-vector in physics is a mathematical tool used to describe physical quantities in four-dimensional spacetime. It consists of four components: three for spatial dimensions and one for time.

## 2. How is four-vector different from regular vector?

Four-vectors are different from regular vectors in that they incorporate the concept of time in addition to the three spatial dimensions. This allows them to describe physical phenomena more accurately in the context of special relativity.

## 3. What is the significance of four-vectors in physics?

Four-vectors are significant in physics because they allow for a unified description of physical quantities in terms of both space and time. They are crucial in special relativity and are used to describe the behavior of particles at high speeds.

## 4. How are four-vectors used in practical applications?

Four-vectors are used in practical applications such as particle physics and cosmology, where the effects of special relativity are significant. They are also used in fields like electromagnetism, where they help describe the behavior of electric and magnetic fields in four-dimensional spacetime.

## 5. Are there any limitations to using four-vectors in physics?

While four-vectors are a valuable tool in physics, they have limitations in certain scenarios. For example, they cannot be used to describe the behavior of objects that are accelerating or changing direction, as these situations require more complex mathematical tools.

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