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stevmg

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This is a continuation of the thread

We have definitions for

1) What is the definition of

2) If

Likewise, proper acceleration or [itex]a_p = a\gamma^3[/itex] also can be [itex] > c [/itex] but I have been told that there is no limit on acceleration other than that the coordinate velocity be [itex] < c [/itex]. Is that valid?

**Derivation of proper time of acceleration in SR**DrGreg said:If you are working in a (t,x) coordinate system, and [itex]\tau[/itex] is proper time:

coordinate acceleration = d^{2}x/dt^{2}

proper acceleration = acceleration measured in the coordinate system of a comoving inertial observer = what an accelerometer measures

rapidity = [tex]\tanh^{-1} \frac {dx/dt}{c}[/tex]

coordinate time =t

coordinate velocity = dx/dt

proper velocity = [tex]dx/d\tau[/tex] although I prefer to call it "celerity" because of possible confusions that can occur (especially over proper acceleration).

yuiop said:[tex]a = \frac{dv}{dt} = \alpha \gamma^{-3} = \alpha (1-v^2/c^2)^{3/2} [/tex]

This can be rearranged to:

[tex]\frac{dt}{dv} = \frac{1}{\alpha(1-v^2/c^2)^{3/2}} [/tex]

Integrating both sides with respect to v gives:

[tex]t= \int \left( \frac{1}{\alpha(1-v^2/c^2)^{3/2}} \right) dv = \frac{v}{\alpha \sqrt{1-v^2/c^2}}[/tex]

When rearranged:

[tex]v= \alpha t \sqrt{1-v^2/c^2} [/tex]

[tex]\rightarrow v^2 = (\alpha t)^2 - (\alpha t v/c)^2[/tex]

[tex]\rightarrow v^2 (1+ (\alpha t /c)^2) = (\alpha t)^2 [/tex]

[tex]\rightarrow v = \frac{\alpha t}{\sqrt{1+(\alpha t/c)^2}} [/tex]

I am still not sure how we get from the [itex]p = mv/\sqrt{1-v^2/c^2}[/itex] and [itex]F=dp/dt[/itex] to [itex]a = \alpha \gamma^3[/itex].

DrGreg said:That's true for the proper time of an inertial object, but in general for a non-inertial object you have to use

[tex]\tau = \int d\tau = \int \sqrt{dt^2 - dx^2 / c^2}[/tex]

from which you get

[tex]\frac{d\tau}{dt} = \sqrt{1 - \frac{1}{c^2}\,\left(\frac {dx}{dt}\right)^2} = \frac{1}{\gamma}[/tex]

i.e.

[tex] \frac{dt}{d\tau} = \gamma[/tex]

hence

[tex] \frac{dx}{d\tau} = \frac{dx}{dt} \, \frac{dt}{d\tau} = \gamma \frac{dx}{dt} = \gamma v[/tex]

in the usual notation.

proper acceleration = the coordinate accelerationmeasured bya comoving inertial observer

JesseM said:Physically the "proper time" along a given worldline just means the time elapsed on a clock that travels along that worldline. If you want to know the proper time between two events on the worldline of a clock moving at constant velocity along the x-axis of some frame, it's just [tex]\Delta \tau = \sqrt{\Delta t^2 - (1/c^2) \Delta x^2 }[/tex], where [tex]\Delta t[/tex] is the difference in coordinate time between the two events and [tex]\Delta x[/tex] is the difference in coordinate position. This can be rewritten as [tex]\Delta \tau = \Delta t \sqrt{1 - (1/c^2) (\Delta x / \Delta t )^2 } = \Delta t \sqrt{1 - v^2 /c^2 }[/tex] which is just the standard time dilation formula. For a worldline where the velocity is varying in a continuous way, you canapproximateits worldline by a "polygonal" worldline made up of a series of constant-velocity segments each lasting a time interval of [tex]\Delta t[/tex] with instantaneous accelerations between them, then if the velocity during the first time interval v_{1}(which could just be the average velocity during the same interval for the path with continuously changing velocity that you are trying to approximate), the second has velocity v_{2}, and the final segment has velocity v_{N}, then the total elapsed time would just be the sum of the elapsed time on each segment, or [tex]\Delta t \sqrt{1 - v_1^2 /c^2} + \Delta t \sqrt{1 - v_2^2 / c^2} + ... + \Delta t \sqrt{1 - v_N^2 /c^2}[/tex]. And in the limit as as the time interval of each segment goes to zero (so the number of segments approaches infinity), this approximation should approach perfect agreement with the proper time on the original path with continuously-changing velocity. Since an integral is just the limiting case of a sum with smaller and smaller intervals (or an infinite series of intervals which each have an 'infinitesimal' time dt), that means the proper time along a worldline where the velocity as a function of time is given by v(t) can always be computed according to the integral [tex]\int_{t_0}^{t_1} \sqrt{1 - v(t)^2 /c^2} \, dt[/tex]

starthaus said:From the blog:

1. Proper speed

[tex]v_p=\frac{dx}{d\tau}=\gamma \frac{dx}{dt}=\gamma v[/tex]

2. Proper acceleration

[tex]a_p=c \frac{d\phi}{d\tau}=\gamma^3*\frac{dv}{dt}=\gamma^3*\frac{d^2 x}{dt^2}=\gamma^3 a[/tex]

t=coordinate time

[itex]\tau[/itex]=proper time

v=coordinate speed

a=coordinate acceleration

We have definitions for

*proper time, proper speed, proper acceleration, coordinate speed, coordinate time, and coordinate acceleration.*1) What is the definition of

*proper distance?*2) If

*proper speed*(or velocity) is [itex] v_p = v* \gamma[/itex] then it is conceivable for [itex] v_p [/itex] to exceed the speed of light. Is that true? If so, what physical meaning does that have? Is there a limit on [itex] v_p < c?[/itex]. Seems like there should be in the sense that there should be*FR for a given observation in which [itex] v \geq c [/itex]***NO**Likewise, proper acceleration or [itex]a_p = a\gamma^3[/itex] also can be [itex] > c [/itex] but I have been told that there is no limit on acceleration other than that the coordinate velocity be [itex] < c [/itex]. Is that valid?

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