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Cleonis
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In another thread a discussion arose about the interpretation of derivatives. For example acceleration is the time derivative of velocity.
Obviously, in the strictly mathematical sense: in relativistic physics and classical physics alike acceleration is the time derivative of velocity.
Let me first discuss an example where both in the mathematical sense and in the sense of physical interpretation there is a process of derivation.
Let there be two coils of conducting wire, I will refer to them as the 'primary coil' and the 'secondary coil'.
Electrically charged particles (electrons) are located in the conductors. As we know, the electrical counterpart of velocity is current. Current is the first time derivative of charge position. The electrical counterpart of acceleration is change of current strength.
If there is change of current strength in the primary coil a current is induced in the secondary coil.
Now, what can you infer when you observe:
- the presence of the primary coil (but you cannot directly observe whether the primary coil even consists of conducting wire.)
- a sinusoidal alternating current in the secondary coil.
Observing what current is induced in the secondary coil is highly informative; it allows you to reconstruct with high fidelity what the current is in the primary coil.
In turn, inferring the existence of current in the primary is immediate proof that the primary consists of a conducting wire.
My point is: the reason you can make those inferences is that change in current strength is physically a derivative process, and the mathematical operation reflects a physical dependency. If a change of current strength exists then a current must exist also, and if there is current then free-to-flow electric charge must be present in the first place.
Or take the example of a single coil, with self-induction. Then there will be inductance at play. When you apply an electromotive force a current will tend to start, but immediately the self-induction kicks in. The change in current strength induces a changing electromagnetic field, which opposes the change in current strength. Inductance is analogous to inertia; change in current strength is proportional to the applied electromotive force.
There is a Newtonian interpretation of dynamics that can be seen as a one-on-one analogy with inductance. According to this interpretation:
'If objects accelerate with respect to the absolute space then velocity with respect to the absolute space must exist also, and if there is velocity relative to the absolute space then absolute position with respect to the absolute space must exist also.'
Relativistic theories of motion affirm the existence of acceleration with respect to the background structure, but as a matter of principle velocity with respect to the background structure does not exist in a relativistic theory.
(Many names for the background structure are in circulation: some refer to it as 'Minkowski spacetime', some refer to it as 'an inertial frame of reference', some refer to it as 'inertial space', etc. Since a background structure exists anyway the most neutral name seems just that: 'the background structure'.)
As a mathematical operation, acceleration is defined as the time derivative of velocity, but in relativistic theories this does not reflect a physical dependency.
Cleonis
JesseM said:[...] coordinate velocity is the derivative of coordinate position, and coordinate acceleration is the derivative of coordinate velocity (and proper acceleration at a given point on an object's worldline is just the coordinate acceleration in the object's instantaneous inertial rest frame at that point, which will match the reading on a co-moving accelerometer at that point). Do you agree that in the mathematical sense this is as true in relativity as it is in classical physics? If you do agree, are you saying that even though it's true mathematically, it's not true "in physical interpretation" in relativity? [...]
Obviously, in the strictly mathematical sense: in relativistic physics and classical physics alike acceleration is the time derivative of velocity.
Let me first discuss an example where both in the mathematical sense and in the sense of physical interpretation there is a process of derivation.
Let there be two coils of conducting wire, I will refer to them as the 'primary coil' and the 'secondary coil'.
Electrically charged particles (electrons) are located in the conductors. As we know, the electrical counterpart of velocity is current. Current is the first time derivative of charge position. The electrical counterpart of acceleration is change of current strength.
If there is change of current strength in the primary coil a current is induced in the secondary coil.
Now, what can you infer when you observe:
- the presence of the primary coil (but you cannot directly observe whether the primary coil even consists of conducting wire.)
- a sinusoidal alternating current in the secondary coil.
Observing what current is induced in the secondary coil is highly informative; it allows you to reconstruct with high fidelity what the current is in the primary coil.
In turn, inferring the existence of current in the primary is immediate proof that the primary consists of a conducting wire.
My point is: the reason you can make those inferences is that change in current strength is physically a derivative process, and the mathematical operation reflects a physical dependency. If a change of current strength exists then a current must exist also, and if there is current then free-to-flow electric charge must be present in the first place.
Or take the example of a single coil, with self-induction. Then there will be inductance at play. When you apply an electromotive force a current will tend to start, but immediately the self-induction kicks in. The change in current strength induces a changing electromagnetic field, which opposes the change in current strength. Inductance is analogous to inertia; change in current strength is proportional to the applied electromotive force.
There is a Newtonian interpretation of dynamics that can be seen as a one-on-one analogy with inductance. According to this interpretation:
'If objects accelerate with respect to the absolute space then velocity with respect to the absolute space must exist also, and if there is velocity relative to the absolute space then absolute position with respect to the absolute space must exist also.'
Relativistic theories of motion affirm the existence of acceleration with respect to the background structure, but as a matter of principle velocity with respect to the background structure does not exist in a relativistic theory.
(Many names for the background structure are in circulation: some refer to it as 'Minkowski spacetime', some refer to it as 'an inertial frame of reference', some refer to it as 'inertial space', etc. Since a background structure exists anyway the most neutral name seems just that: 'the background structure'.)
As a mathematical operation, acceleration is defined as the time derivative of velocity, but in relativistic theories this does not reflect a physical dependency.
Cleonis