Proper acceleration and g-forces

[granpa]

proper acceleration is the acceleration actually felt/experienced by an accelerating body. the integral of this gives the proper velocity. this makes intuitive sense. but the derivation of the formula for proper acceleration doesn't make intuitive sense to me. I'm very confused about how one would determine how many g-forces were being experienced by an astronaut moving very near the speed of light.

 

[JustinLevy]

Since you say you've already seen derivations and it wasn't helpful, can you please explain a bit more about your current level of understanding? It is easier to figure out where a flaw in understanding is if you explain what you do know and your confusion the best you can.

In case the problem is on the level of definitions, I'll just give a short bit here:
One things that should be kept in mind is that the derivatives you refer to are with respect to proper time of the object.

If you have the description of an object's motion according to an inertial frame, first rewrite this to be a function of proper time instead of the coordinate time. Then take the second derivative with respect to the proper time ... and that's it! If you want to know what the components look like for the astronaut, just transform your answer to the instantaneous inertial frame of the astronaut.

 

[granpa]

thats just it. the wikipedia article differentiated with respect to 'map time' (whatever that is) not proper time.

I guess its better to start with velocity than to jump to acceleration. if rocket A is moving at gamma=g and rocket B is moving at gamma=g+delta g (or rather g+dg) then what is the velocity of rocket B as measured by rocket A? time dilation and length contraction are easy but what about simultaneity?

 

[serakac]

The derivatives you refer are with respect to proper time of the object. Proper acceleration is the physical acceleration experienced by an object. It equals the coordinate acceleration if you are using an inertial coordinate system in flat spacetime, provided the object's proper-velocity. (momentum per unit mass) is much less than lightspeed.
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[granpa]

http://en.wikipedia.org/wiki/Proper_velocity#Unidirectional_acceleration_via_proper_velocity

(coordinate velocity v=dx/dt, proper-velocity w=dx/dτ, and Lorentz factor γ=dt/dτ)

In flat spacetime, proper-velocity is the ratio between distance traveled relative to a reference map-frame (used to define simultaneity) and proper time τ elapsed on the clocks of the traveling object. It equals the object's momentum p divided by its rest mass m, and is made up of the space-like components of the object's four-vector velocity.

In flat spacetime, proper acceleration is the three-vector acceleration experienced in the instantaneously-varying frame of an accelerated object[11]. Its magnitude α is the frame-invariant magnitude of that object's four-acceleration. Proper-acceleration is also useful from the vantage point (or spacetime slice) of an observer. Not only may observers in all frames agree on its magnitude, but it also measures the extent to which an accelerating rocket "has its pedal to the metal".
In the unidirectional case i.e. when the object's acceleration is parallel or anti-parallel to its velocity in the spacetime slice of the observer, the change in proper-velocity is the integral of proper acceleration over map-time i.e. Δw=αΔt for constant α. At low speeds this reduces to the well-known relation between coordinate velocity and coordinate acceleration times map-time, i.e. Δv=aΔt.

By grouping γ with v in the expression for relativistic momentum p, proper velocity also extends the Newtonian form of momentum as mass times velocity to high speeds without a need for relativistic mas

 

[granpa]

http://en.wikipedia.org/wiki/Four-vector

When considering physical phenomena, differential equations arise naturally; however, when considering space and time derivatives of functions, it is unclear which reference frame these derivatives are taken with respect to. It is agreed that time derivatives are taken with respect to the proper time (τ) in the given reference frame. It is then important to find a relation between this time derivative and another time derivative (taken in another inertial reference frame). This relation is provided by the time transformation in the Lorentz transformations and is:
dtau/dt=1/gamma
where γ is the Lorentz factor. Important four-vectors in relativity theory can now be defined, such as the four-velocity of an x(tau) world line is defined by:
u=dxdt/dtdtau
The four-acceleration is given by:
du/dtau