Okay, I can do that. What were the initial conditions on those 2 eqns? That's the last thing I need to know. Did you take the initial conditions to be from the view of the accelerated frame? That is, Xo', To', Vo'? If I invert the equations to get x', ct' as functions of x, ct, how will the initial conditions transform? Is there a transformation for Xo,To,Vo to Xo', To', Vo'?
@natebezayiff My initial conditions were simply that I let the origins of the two frames coincide at time zero. In doing the antiderivatives you have integration constants that correspond to where you want the origins to be to start.
@natebezayiff Also I should point out that the case of constant proper acceleration is not the same thing as constant coordinate acceleration which is actually a mess. The case of constant coordinate acceleration is also only valid of course as long as
Could I do it for constant acceleration in the x-direction (for all frames) in the proper frame? Would it be as easy as treating your bottom two equations for ct,x as a system of 2 eqns with 2 unknowns (the same way the constant vel Lor Trans are done) where x' and ct' are the unknowns?
@natebezayiff Thats how you invert them, yes. The case of constant proper acceleration is pretty easy to invert. The results for the constant proper acceleration case is the last two equations at the very end of the video. To invert that case, you'll need the hyperbolic trig identity
Also, do you have the inverse lorentz transformations for the accelerated primed frame in terms of the unaccelerated unprimed frame? That is, do you have x' and ct' as functions of x, ct ? Would the initial conditions be frame dependent: X=Xo, T=To, V=Vo or X'=X'o, T'=T'o, V'=V'o? Sorry for all the posts, this is something I am really interested and not related to HW or anything like that, but is hard to find in txtbooks or on the web.
@natebezayiff You won't find it because they don't know them yet. There doesn't exist a straight forward set of inverse equations. Its something like trying to solve for x in
y = x + lnx
You can find the value of x for any corresponding y so the inversion can be said to exist, but its not a simple function.
What were the initial conditions in the last steps of your proof (Bottom expressions of ct and x at time 8:23) ? I'll assume X'=0, V'=0, T'=0? Does the above expressions handle completely general intitial conditions: X'o, T'o, V'o (all X-dir) where these initial parameters are non-zero?
@natebezayiff Yes the conditions I used was that the origins meet with zero velocity at time zero. Yes my transformation equations can apply arbitrary initial conditions.
Okay, I can do that. What were the initial conditions on those 2 eqns? That's the last thing I need to know. Did you take the initial conditions to be from the view of the accelerated frame? That is, Xo', To', Vo'? If I invert the equations to get x', ct' as functions of x, ct, how will the initial conditions transform? Is there a transformation for Xo,To,Vo to Xo', To', Vo'?
natebezayiff 3 months ago
@natebezayiff My initial conditions were simply that I let the origins of the two frames coincide at time zero. In doing the antiderivatives you have integration constants that correspond to where you want the origins to be to start.
WaiteDavidMSPhysics 3 months ago
@natebezayiff Oh and yes, that the velocity at time zero was zero.
WaiteDavidMSPhysics 3 months ago
@natebezayiff Also I should point out that the case of constant proper acceleration is not the same thing as constant coordinate acceleration which is actually a mess. The case of constant coordinate acceleration is also only valid of course as long as
beta < 1
WaiteDavidMSPhysics 3 months ago
Could I do it for constant acceleration in the x-direction (for all frames) in the proper frame? Would it be as easy as treating your bottom two equations for ct,x as a system of 2 eqns with 2 unknowns (the same way the constant vel Lor Trans are done) where x' and ct' are the unknowns?
natebezayiff 3 months ago
@natebezayiff Thats how you invert them, yes. The case of constant proper acceleration is pretty easy to invert. The results for the constant proper acceleration case is the last two equations at the very end of the video. To invert that case, you'll need the hyperbolic trig identity
cosh2(u) - sinh2(u) = 1
WaiteDavidMSPhysics 3 months ago
Also, do you have the inverse lorentz transformations for the accelerated primed frame in terms of the unaccelerated unprimed frame? That is, do you have x' and ct' as functions of x, ct ? Would the initial conditions be frame dependent: X=Xo, T=To, V=Vo or X'=X'o, T'=T'o, V'=V'o? Sorry for all the posts, this is something I am really interested and not related to HW or anything like that, but is hard to find in txtbooks or on the web.
natebezayiff 3 months ago
@natebezayiff You won't find it because they don't know them yet. There doesn't exist a straight forward set of inverse equations. Its something like trying to solve for x in
y = x + lnx
You can find the value of x for any corresponding y so the inversion can be said to exist, but its not a simple function.
WaiteDavidMSPhysics 3 months ago
@natebezayiff I should add that you can find simple inversions for particular cases of acceleration though.
WaiteDavidMSPhysics 3 months ago
What were the initial conditions in the last steps of your proof (Bottom expressions of ct and x at time 8:23) ? I'll assume X'=0, V'=0, T'=0? Does the above expressions handle completely general intitial conditions: X'o, T'o, V'o (all X-dir) where these initial parameters are non-zero?
natebezayiff 3 months ago
@natebezayiff Yes the conditions I used was that the origins meet with zero velocity at time zero. Yes my transformation equations can apply arbitrary initial conditions.
WaiteDavidMSPhysics 3 months ago
The coordinate of an event according to the inertial frame is
(ct,x,y,z)
The coordinate of the event according to my coordinates for an accelerated observer is
(ct',x',y',z')
So, yes its the x' axis part of the position of the event according to the accelerated observer.
WaiteDavidMSPhysics 3 months ago
What does X' mean? Is this the position of an event as measured in the frame of the accelerated observer?
natebezayiff 3 months ago
holy shit
TheMarsVo1ta 1 year ago