Lorentz transformation

[Tim_Lou]

i understand the lorenz transformation in "one direction" x. but what about someting like a uniform circular motion or other accelerated motion? does the lorenz transformation "in one direction" always hold true in the differiential level even with acceleration? if so then does:

[math]dt'={(dt-dxv/c^2)\over{\gamma}}[/math]

 

always hold true even though there is acceleration? if not is it a good approximation though?

 

it seems that i have to understand this equation in order to advance to other topics... any help is appreciated.

 

i just need a quick answer on whether not it holds true with acceleration. people always tell me that i need general relativity.... well i just wanna know if i can apply this equation without all the glory of Einstein's field equatiosn and stuffs.

[IDMclean]

x holds constant.

I would say intuitively that it would need to hold across acceleration, as lorentzian transformation is ment to maintain the difference of states across a relative set of frames... I think.

[Erasmus00]

i understand the lorenz transformation in "one direction" x. but what about someting like a uniform circular motion or other accelerated motion? does the lorenz transformation "in one direction" always hold true in the differiential level even with acceleration? if so then does:

[math]dt'={(dt-dxv/c^2)\over{\gamma}}[/math]

 

always hold true even though there is acceleration? if not is it a good approximation though?

 

it seems that i have to understand this equation in order to advance to other topics... any help is appreciated.

 

i just need a quick answer on whether not it holds true with acceleration. people always tell me that i need general relativity.... well i just wanna know if i can apply this equation without all the glory of Einstein's field equatiosn and stuffs.

 

 

You can always describe a particle as being in an "instantaneous inertial frame" and so you can use lorentz transformations to move into that one frame. However, it only holds for an infinitesimal element of "length" along an accelerated curve in minkowski space (world line). The trick then is just to integrate along the accelerated world line.

 

As long as you do calculations from within an inertial frame, you can use SR to describe accelerations. See any SR book for a discussion of 4 vectors (including 4 acceleration).

-Will

[Tim_Lou]

so a fram with zero velocity (initially) but a non-zero acceleration will not experience any time dilation (relative to an inertial frame) according to lorenz transformation and hence same for gravity. however that is not correct in GR. is this one of the situation where the equation breaks down?

[Erasmus00]

so a fram with zero velocity (initially) but a non-zero acceleration will not experience any time dilation according to lorenz transformation and hence same for gravity. however that is not correct in GR. is this one of the situation where the equation breaks down?

 

We need two frames to define any kind of time dilation. (time dilation with relation to what?)

 

A frame with the same velocity as your own (i.e. 0 relative velocity) will, at that instant, exhibit no time dilation according to your own clock. This is true in any flat space, no matter if you use SR or GR to analyze the situation.

-Will

[Tim_Lou]

but let's say relative to my frame of reference (I'm not being accelerated), a clock is being accelerated from rest relative to me. now according to SR, the clock shows no time dilation at the instant when the clock is at rest. However, let's say the clock is accelerated due to gravity, according to GR, there should be some form of space-time distortion regardless of the clock's velocity; contradictory to the prediction of SR, where dx=dx',dy=dy',dz=dz',dt=dt' at the instant the clock is at rest. Am I thinking something wrong here?

[IDMclean]

A body is always at rest relative to itself.

[Erasmus00]

but let's say relative to my frame of reference (I'm not being accelerated), a clock is being accelerated from rest relative to me. now according to SR, the clock shows no time dilation at the instant when the clock is at rest. However, let's say the clock is accelerated due to gravity, according to GR, there should be some form of space-time distortion regardless of the clock's velocity; contradictory to the prediction of SR, where dx=dx',dy=dy',dz=dz',dt=dt' at the instant the clock is at rest. Am I thinking something wrong here?

 

Accelerating due to gravity is fundamentally different then accelerating in flat space. What you are missing is that the relationship

 

[math]d\tau^2 =dt^2-dx^2-dy^2-dz^2[/math]

 

only holds in flat space. You cannot use SR in a gravitational field. You can use SR when things accelerate within flat space.

-Will

[Tim_Lou]

so, no gravity, no equivalent principle in SR. As long as the acceleration is not insanely high, i guess this is a very good approximation. thx for the helps and clarifications.

 

now time to try to derive all sorts of "lorenz transformation" in different situation and calculate time dilation for twin's paradox with acceleration :beer::confused:

[Erasmus00]

now time to try to derive all sorts of "lorenz transformation" in different situation and calculate time dilation for twin's paradox with acceleration :beer::confused:

 

Time dilation in the twin's paradox isn't to bad. Just do all your calculations from the inertial frame, and integrate along the accelerating twin's world line.

-Will