SR w/ Acceleration: Distance Measured by A

In summary, the conversation discusses the problem of finding the distance between two objects, A and B, in a given time and frame. The solution presented involves using the invariance of vectors and calculating the distance as measured by A in the primed frame. However, the individual questioning this solution raises concerns about the use of the temporal distance and suggests finding a vector with a zero temporal distance instead.
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Gaussian97
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Homework Statement
Let Alice and Bob be accelerated in Charles RF, with world lines with respect to C given by (1), what is the distance between A and B measured from C? What is the distance of A and B measured from A?
Relevant Equations
$$c = 1$$
$$t_{AB}=\frac{\sinh{(a\tau_{AB})}}{a}, \qquad x_{AB}=x_{0,AB}+\frac{\cosh{(a\tau_{AB})}-1}{a} \qquad (1)$$
Well, using (1) is easy to see that, at a given time in C ##t## both curves are described with the same value of ##\tau##, i.e. ##\tau_A=\tau_B=\tau##. So the corresponding positions at a given time ##t## are
$$x_{AB}=x_{0,AB}+\frac{\cosh{(a\tau)}-1}{a}$$
and therefore
$$\Delta x \equiv x_B - x_A = x_{0,B}+\frac{\cosh{(a\tau)}-1}{a} - x_{0,A}-\frac{\cosh{(a\tau)}-1}{a} = x_{0,B} - x_{0, A}$$
so A and B are always at the same distance from each other. No big problem here (I think)
My problem comes when I have to compute the distance measured in A.

The solution that I have says that the vector the vectors are invariant (which I agree) and therefore
$$\Delta x \vec{e}_x = \Delta x' \vec{e}'_x +\Delta t' \vec{e}'_t\Longrightarrow \Delta x' = \Delta x \vec{e}_x\cdot \vec{e}'_x = \cosh{(a\tau)} \Delta x$$
But I don't understand why this is the distance measured by A, because the "temporal distance" here is not zero, shouldn't we find a vector with ##\Delta t'=0## and then compute the distance?
 
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  • #2
Gaussian97 said:
The solution that I have says that the vector the vectors are invariant (which I agree) and therefore
$$\Delta x \vec{e}_x = \Delta x' \vec{e}'_x +\Delta t' \vec{e}'_t\Longrightarrow \Delta x' = \Delta x \vec{e}_x\cdot \vec{e}'_x = \cosh{(a\tau)} \Delta x$$
But I don't understand why this is the distance measured by A, because the "temporal distance" here is not zero, shouldn't we find a vector with ##\Delta t'=0## and then compute the distance?
I agree with you. I'm not sure what they're doing in the solution that you posted. I take it that the primed frame is an inertial reference frame for which A is instantaneously at rest at the instant that A determines the distance to B. The distance between A and B, as measured by A, would be determined from simultaneous measurements of the positions of A and B in the primed frame (at the instant that A is at rest in this frame). So, as you say, ##\Delta t'=0##. The corresponding ##\Delta t## would not be zero.
 
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Related to SR w/ Acceleration: Distance Measured by A

1. What is special relativity with acceleration?

Special relativity with acceleration is a theory that combines the principles of special relativity (which deals with the behavior of objects in uniform motion) with the effects of acceleration. It describes how objects behave when they are accelerating, which is when their velocity is changing.

2. How is distance measured in special relativity with acceleration?

In special relativity with acceleration, distance is measured using the concept of "proper distance". This is the distance between two points in space as measured by an observer who is at rest relative to those points. It takes into account the effects of time dilation and length contraction.

3. How does acceleration affect time in special relativity?

In special relativity, acceleration affects time in two ways. First, it causes time dilation, which means that time appears to pass slower for an accelerating object compared to an object at rest. Second, it causes the phenomenon of "relativity of simultaneity", where events that are simultaneous for an observer at rest may not be simultaneous for an accelerating observer.

4. Can an object travel faster than the speed of light in special relativity with acceleration?

No, according to the principles of special relativity, the speed of light is the maximum speed at which any object can travel. This applies to objects that are accelerating as well. As an object approaches the speed of light, its mass increases and it requires an infinite amount of energy to accelerate it further.

5. How is energy related to acceleration in special relativity?

In special relativity, energy is related to acceleration through the famous equation E=mc². This means that an object's energy is directly proportional to its mass and the square of its velocity. As an object's velocity increases, its energy also increases, and this is why it requires an infinite amount of energy to accelerate an object to the speed of light.

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