- #1

member 728827

- TL;DR Summary
- Can Born rigidity condition be used at not relativistic speeds, and if so, can we use thereafter the Newton's laws with the reached Born acceleration, and do Energy before/after balances in the Classical way?

Hello,

I try to better understand how and when I can apply the Born rigidity condition.

So, for the following example:

We've two space probes (P

For our experiment, we need the proper distance to keep exactly constant as observed by P

The speed, relative to Earth-frame stationary starting position, is about 10 Km/s at t

The questions are:

- Is the

If the Born condition holds at this non-relativistic speeds,

##\alpha_{bbo} = \frac{c^2 \cdot \alpha_a}{c^2 + \alpha_a \cdot L_{ba0}'}##

then

- After the Born rigidity condition has been reach, can we apply locally Newton's laws to P

##M \cdot (\alpha_b - \alpha_{bbo}) = F_{ion}##

- After the Born rigidity condition has been reach, do the following Energy balance between just before and just after the Born condition be considered locally at P

##\frac{1}{2} \cdot M \cdot v_{ba}^2 = \int_0^{\delta} F'_{rel} \cdot d\delta - \int_0^{\delta} F'_{ion} \cdot d\delta##

where v

I think that's mostly Ok, and there's no problem doing that, but...?. Any help or suggestions will be appreciated.

Thanks

I try to better understand how and when I can apply the Born rigidity condition.

So, for the following example:

We've two space probes (P

_{a}and P_{b}), that travel at an**exact equal and same proper acceleration**. At a given time t_{b0}in P_{b}, and as measured by P_{b}, the distance is L_{ba0}(it's quite large).For our experiment, we need the proper distance to keep exactly constant as observed by P

_{a}/P_{b}, so a very precise ion thruster starts in P_{b}at t_{b0}, and adjusts the α_{b}proper acceleration to the required value to the Born rigidity condition. It has enough precision to do that (even at the pico-Newton scale).The speed, relative to Earth-frame stationary starting position, is about 10 Km/s at t

_{b0}(in S_{b}). It's the order of magnitude what's important. The L_{ba0}length exact value doesn't matter either, but what's critical is that the final proper distance after going to Born rigidity condition keeps constant thereafter, or our experiment would not work reliably.The questions are:

- Is the

**Born rigidity**condition applicable in situations where the speeds are clearly not relativistic, or that equation only holds when talking about relativistic speeds. If so, what is the minimum v/c ratio to use it?If the Born condition holds at this non-relativistic speeds,

##\alpha_{bbo} = \frac{c^2 \cdot \alpha_a}{c^2 + \alpha_a \cdot L_{ba0}'}##

then

- After the Born rigidity condition has been reach, can we apply locally Newton's laws to P

_{b}, and write :##M \cdot (\alpha_b - \alpha_{bbo}) = F_{ion}##

- After the Born rigidity condition has been reach, do the following Energy balance between just before and just after the Born condition be considered locally at P

_{b}, and write:##\frac{1}{2} \cdot M \cdot v_{ba}^2 = \int_0^{\delta} F'_{rel} \cdot d\delta - \int_0^{\delta} F'_{ion} \cdot d\delta##

where v

_{ba}is the speed P_{b}has relative to P_{a}just when the ion-thruster starts and due to the non-simultaneity of accelerations at both ends. F'_{rel}is like a "fictitious" relativistic force, much like the Coriolis force can be for an observer in a non-inertial rotating frame, and δ is the proper length measured in P_{b}, traveled by P_{b}during the transient phase.I think that's mostly Ok, and there's no problem doing that, but...?. Any help or suggestions will be appreciated.

Thanks