Operational definition of four velocity

In summary: An inertial observer who is at rest relative to an accelerating object can measure the time using its clocks and use dt = gamma dtau. Alternatively, time can be measured in the respect of the reference frame comoving with the traveling particle, but this is not an inertial one so more dilemmas arise.

Dear Friends

Bearing in mind the special relativity definition of "observer" (a system of rods and clock disposed everywhere in the space), what's the
operational definition of four velocity (and therefore of four acceleration, four force, etc)?

Obviously, no problems in detecting the displacement's components of the
traveling particle (in the observer's reference frame) but how about the proper time tau? I'm mulling about two possibilities:

(1)
The observer can measure the time using its clocks and then use
dt = gamma dtau
(but I don't like it, because it repose on too theoretical premises and it's no really "experimental fashioned" )

(2)
Time can be measured in the respect of the reference frame comoving with the traveling particle, but this is not an inertial one so more dilemmas
rise up.

Thanks for unscrambling my English.
Have a great day !

Barbara Da Vinci
Rome

What you are really asking for is an operational definition of proper time $\tau$. The proper time of an object is the time recorded by a clock carried by the object. The clock hypothesis postulates that an accelerating clock ticks at the same rate as a co-moving inertial clock. I.e. $dT/d\tau = 1$ where T is the time recorded by an inertial observer who, at that moment, is at rest relative to the accelerating object. Note that proper time is defined only for the object itself, you can't extend it to events that are not on the object's worldline.

That's the operational definition. The mathematical definition is

$$d\tau^2 = dt^2 - \frac{dx^2 + dy^2 + dz^2}{c^2}$$​

where (t, x, y, z) are any inertial coordinates (whether co-moving or not), which is equivalent to your (1).

Either way, the 4-velocity is then

$$\textbf{V} = (dt/d\tau, dx/d\tau, dy/d\tau, dz/d\tau)$$​

I agree, you got fully the main point. Using a metaphor, you are essentially confirming the clock hypothesis to be a sort of relativistic Euclid's fifth postulate, an assumption no way coming out from the standard hypotheses behind the Lorentz transformations. Do I understand it right ?

> an inertial observer who, at that moment, is at rest relative to the accelerating object

Well, it's a widely spread but somewhat paradoxical concept: "at rest" entail an observation over a finite time interval. Thinking the word line approximated by means of a sequence of infinitesimal extended straight lines doesn't sound entirely good to me: measure and integration theory is a subtle and tricky business ...

Best regards!

Barbara Da Vinci
Rome

(http://math.ucr.edu/home/baez/physics/Relativity/SR/clock.html)
what in the following I quote (may be someone is interested ...). Bye.

"The clock postulate is not meant to be obvious, and it can't be proved. It's not merely some kind of trivial result obtained by writing special relativity using non-cartesian coordinates. Rather, it's a statement about the physical world. But we don't know if it's true; it's just a postulate. For instance, we can't magically verify it by noting that the Lorentz transform is only a function of speed, because the Lorentz transform is something that's built before the clock postulate enters the picture. Also, we cannot simply wave our arms and maintain that an acceleration can be treated as a sequence of constant velocities that each exist only for an infinitesimal time interval, for the simple reason that an accelerating body (away from gravity) feels a force, while a constant-velocity body does not. Although the clock postulate does speak in terms of constant velocities and infinitesimal time intervals, there's no a priori reason why that should be meaningful or correct. It's just a postulate! This is just like the fact that even though a 1000-sided polygon looks pretty much like a circle, a small piece of a circle can't always be treated as an infinitesimal straight line: after all, no matter how small the circular arc is, it will always have the same radius of curvature, whereas a straight line has an infinite radius of curvature. It also won't do to simply define a clock to be a device whose timing is unaffected by its acceleration, because it's not clear what such a device has got to do with the real world: that is, how well it approximates the thing we wear on our wrist.

> an inertial observer who, at that moment, is at rest relative to the accelerating object

Well, it's a widely spread but somewhat paradoxical concept: "at rest" entail an observation over a finite time interval. Thinking the word line approximated by means of a sequence of infinitesimal extended straight lines doesn't sound entirely good to me: measure and integration theory is a subtle and tricky business ...
Perhaps it would have been better for me to say "an inertial frame in which the object's velocity ($d\textbf{x}/dt$) is zero at that moment".

Or, to put it another way, the clock hypothesis postulates that whenever, in an inertial frame $(t,x,y,z)$,

$$\frac{dx}{dt} = \frac{dy}{dt} = \frac{dz}{dt} = 0$$...(1)​

then

$$\frac{dt}{d\tau} = 1$$...(2)​

At any event along the object's path you will be able to find an inertial frame satisfying (1) so $d\tau$ can be defined, at that event only, via (2). To calculate $\tau$ along the whole path you have to integrate $d\tau$.

I'm assuming you understand integration and differentiation in calculus.

1. What is an operational definition of four velocity?

An operational definition of four velocity is a mathematical concept used in the field of special relativity to describe the rate of change of an object's position in four-dimensional spacetime. It takes into account both the object's speed and direction of motion.

2. How is four velocity different from regular velocity?

Four velocity is different from regular velocity in that it takes into account the object's motion in all four dimensions of spacetime, whereas regular velocity only considers motion in three dimensions of space. Four velocity also includes information about the object's direction of motion, while regular velocity does not.

3. Can four velocity be negative?

Yes, four velocity can be negative. This would indicate that the object is moving in the opposite direction of the positive direction in the four dimensions of spacetime. However, the magnitude of four velocity is always positive, as it represents the speed of the object regardless of its direction of motion.

4. How is four velocity measured or calculated?

Four velocity is typically measured or calculated using special relativity equations, which take into account the object's speed and direction of motion in four dimensions of spacetime. It can also be determined by taking the derivative of the object's position with respect to proper time, which is a concept in special relativity that takes into account the effects of time dilation and length contraction.

5. What practical applications does four velocity have?

Four velocity has practical applications in the field of physics, particularly in the study of special relativity. It is used to describe the motion of particles and objects traveling at high speeds, such as those found in nuclear reactions or in space. It is also used in the development of theories and technologies related to time travel and faster-than-light travel.