# Special relativity and acceleration

• LCSphysicist
In summary, the conversation discusses a particle moving along the x-axis with a constant acceleration of g in its instantaneous rest frame. The goal is to find x and t as functions of proper time τ, assuming the particle starts at x0 with zero velocity. The conversation also mentions transforming the equations to the rest frame coordinates and potential mistakes in the approach.
LCSphysicist
Homework Statement
N
Relevant Equations
N
A particle is moving along the x-axis. It is uniformly accelerated in the sense
that the acceleration measured in its instantaneous rest frame is always g, a constant.
Find x and t as functions of the proper time τ assuming the particle passes through
x0 at time t = 0 with zero velocity.I

n particle frame, the acceleration is constant and given by g.

So we have $$dv/d\tau = g \implies x = x_o + v_ot + g\tau^2/2$$
Using the initial conditions,$$x = x_o + g\tau^2/2$$

So now we have to transform it to the rest frame coordinates/lab frame.

$$\begin{pmatrix} ct'\\x' \end{pmatrix} = \begin{pmatrix} \gamma & \gamma \beta \\ \gamma \beta & \gamma \end{pmatrix} \begin{pmatrix} ct \\ x = x_o + g\tau^2/2 \end{pmatrix}$$

I am using beta instead of minus beta, because i am changing from a frame in motion to a frame in rest.

Now, assuming that $$(dv/dt) dt/d\tau = g \implies v= gt/\gamma$$
And so, $$\gamma = \sqrt{1+g^2t^2}$$
$$\beta = -gt/(c\gamma)$$

that implies $$t` = \sqrt{1+g^2t^2}(c\tau + \beta*( x_o + g\tau^2/2))/c$$
and $$x' = \sqrt{1+g^2t^2}(c \tau \beta +(x_o + g\tau^2/2))$$

But i am not sure about these results i get, i have the impression i am doing something wrong. Is it right?

It might be helpful to see the form of the equations when $\beta$ is small. (Correspondence principle.)

robphy said:
It might be helpful to see the form of the equations when $\beta$ is small. (Correspondence principle.)
I think i could have had a mistake when i integrated the v, reading it again... But, i think what's matter more to me is if the approach is right. I mean, solve it using the proper time, and so changing to the rest frame using the inverse matrix transform, what do you think?

You only have ##dx/d\tau = g## instantaneously in any frame when the particle is at rest in that frame. As soon as the particle deviates from zero velocity in that frame, it is no longer valid. Therefore you willhave to rethink your approach.

## 1. What is special relativity?

Special relativity is a theory developed by Albert Einstein that explains how the laws of physics are the same for all observers in uniform motion, regardless of their relative velocities. It also introduces the concept of space-time, where space and time are intertwined and can be affected by the presence of massive objects.

## 2. How does acceleration affect special relativity?

According to special relativity, acceleration can cause time dilation, where time appears to pass slower for an observer in motion compared to an observer at rest. This is due to the increase in velocity and the resulting changes in space-time. Additionally, acceleration can also cause length contraction, where objects appear shorter in the direction of motion.

## 3. What is the difference between special relativity and general relativity?

Special relativity deals with the laws of physics in inertial (non-accelerating) reference frames, while general relativity extends these laws to include acceleration and gravity. General relativity also introduces the concept of curved space-time, where the presence of massive objects can cause distortions in space and time.

## 4. Can special relativity explain faster-than-light travel?

No, according to special relativity, the speed of light is the maximum speed at which any object can travel. As an object approaches the speed of light, its mass increases and it requires an infinite amount of energy to accelerate it further. Therefore, it is impossible for any object to travel faster than the speed of light.

## 5. How is special relativity applied in everyday life?

Special relativity has many practical applications in areas such as GPS technology, particle accelerators, and nuclear power plants. It also plays a crucial role in our understanding of the universe, including phenomena such as black holes and the expansion of the universe.

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