# Relativistic Force and Acceleration: Understanding the Relationship

• quasar987
In summary, the relativistic law of motion states that the force acting on a particle is equal to the change in momentum over time, which takes into account the particle's mass, velocity, and acceleration. However, the direction of acceleration may not be the same as the force, as it is influenced by the particle's speed. This makes it difficult to use acceleration as a reliable measure in special relativity.
quasar987
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The relativistic law of motion is

$$\vec{F} = \frac{d\vec{p}}{dt} = \frac{d}{dt}\frac{m\vec{u}}{\sqrt{1-(u/c)^2}} = \frac{m}{\sqrt{1-(u/c)^2}}\frac{d\vec{u}}{dt}+\frac{1}{c^2}\frac{m\vec{u}}{(1-(u/c)^2)^{3/2}}\frac{du}{dt} = \frac{m}{\sqrt{1-(u/c)^2}}\vec{a} + \frac{a}{c^2}\frac{m\vec{u}}{(1-(u/c)^2)^{3/2}}$$

So this means that the acceleration of the particle does no take place in the same diretion as the force. Instead it is in some weird direction dictated by the actual speed of the particle.

Is this right?

Yes,but it can be shown (actually by a definition) that the 4 force & the 4 acceleration are collinear

$$f^{\mu}=:m_{0}w^{\mu}$$

,where,obviously,$c=1$.

Daniel.

quasar,

Yeah, quite a mess isn't it! There's a special case or two (force parallel to v, is one of them I think), but generally, trying to use F=dp/dt to find the acceleration of a particle moving at high speed is not fun. Fortunately (as Dextercioby said) you don't have to.

As Anthony French says in his intro to SR after deriving the transforms for acceleration, "The main lesson to be learned from the above calculations is that acceleration is a quantity of limited and questionable value in SR...Certainly the proud position that it holds in Newtonian dynamics has no counterpart here."

Very interesting!

## 1. What is the Relativistic Law of Motion?

The Relativistic Law of Motion, also known as the special theory of relativity, is a scientific theory that explains the relationship between space and time and how they are affected by the motion of objects. It was first developed by Albert Einstein in 1905 and has since been confirmed by numerous experiments and observations.

## 2. How does the Relativistic Law of Motion differ from classical mechanics?

The Relativistic Law of Motion differs from classical mechanics in a few key ways. Firstly, it takes into account the constant speed of light and the fact that it is the same for all observers. Secondly, it considers time and space as relative concepts that are affected by the motion of objects. Lastly, it predicts that the laws of physics should be the same for all observers, regardless of their relative motion.

## 3. What are the implications of the Relativistic Law of Motion?

The Relativistic Law of Motion has several implications, including the concept of time dilation, where time appears to pass slower for objects moving at high speeds. It also explains the phenomenon of length contraction, where objects appear to shrink in the direction of their motion. Additionally, it provides the framework for understanding the relationship between energy and mass, as described by Einstein's famous equation, E=mc^2.

## 4. How has the Relativistic Law of Motion been confirmed?

The Relativistic Law of Motion has been confirmed through various experiments and observations, such as the Michelson-Morley experiment, which showed that the speed of light is constant regardless of the observer's motion. Other experiments, such as the Hafele-Keating experiment, have also confirmed the predictions of time dilation and length contraction. Furthermore, the theory has been used to make accurate predictions in fields such as particle physics and cosmology.

## 5. How does the Relativistic Law of Motion impact our daily lives?

Although the effects of the Relativistic Law of Motion may not be noticeable in our daily lives, it has had a significant impact on modern technology. For example, the Global Positioning System (GPS) relies on the theory to accurately determine the positions of satellites and receivers on Earth. Additionally, the theory has led to advancements in fields such as nuclear energy and particle accelerators, further shaping our modern world.

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