# Why position and velocity are symmetries and acceleration is not?

Position and velocity are symmetries. The law of physics do not change if the observer changes his position or velocity.

But acceleration which is just a derivative of velocity is not a symmetry. In other words, unlike velocity,acceleration does not change if we change the reference frame.

What is the reason for this difference?

• Because Newton's first law – Kenshin Feb 24 '17 at 8:56
• Because we don't have the ultimate theory yet, in which all observers should be equal. – velut luna Feb 24 '17 at 9:00

I think you are asking more than one question here.

The 1st is Why are the Laws of Physics the same in all inertial reference frames? Ultimately this question can be reduced to Why is the speed of light the same in all inertial reference frames? This is an observational fact for which there is no accepted theory to explain it. It is a postulate of Special Relativity Theory so we cannot expect that theory to provide any explanation.

The 2nd question is much easier to answer. It is asking why position and velocity are different but acceleration is the same in different inertial reference frames.

This is an extension of the fact that 2 frames which have different origins but no relative motion will measure a different position but the same velocity, acceleration, etc, for an object. And 2 frames with a constant relative velocity will measure different velocities as well as positions, but the same acceleration, etc.

Acceleration is the same in all inertial reference frames simply because there is no relative acceleration between them. If the co-ordinates of an event in frames S and S' are related by $x'=x+vt$ then $\ddot x'=\ddot x$, ie acceleration - and all higher derivatives - are the same.

Acceleration is not the same if measured in different reference frames if there is relative acceleration between them. If co-ordinates in S, S' are related by $x'=x+vt+\frac12at^2$ then $\ddot x'=\ddot x + a$. However, if 2 frames have the same acceleration, and differ only in the position and velocity of the co-ordinate axes, they will agree on the acceleration measured for an object.

In Newtonian mechanics and special relativity the main concept is "inertial frame". One can define the class of equivalence of frames called inertial as those that moves relatively to each other with constant velocity. In all of this frames, the equations of physics are invariant (covariant).

First: The definition of inertial frame of reference looks like a tautological one: "One frame is inertial if moves at constant velocity relatively to an inertial frame"; but in fact is not, and the concept is subtle: One choose from the particles that can be seen one with certein characteristic and call it "free particle" which is an inertial frame (by definition), develop the phisics of this free particle and extend it to all inertial frames.

Second: The way of extending this laws of physics to inertial frame relies on the transformations of the coordinates. Galilean and Lorentz transformations take in account only position and velocity, and it is expected that any other reparametritation of velocity and position do not affect the laws we have developed.

The key is the way one define that transformations. Let put it a bit mor formal: One must define the space one is using; in Newtonian mechanics is a 3d-space manifold with an ansolute time (a scalar function defined in the whole space whose gradient is non vanishing) and an euclidean metric, while in special relativity is a 4d-spacetime manifold endowed with Minkowsky metric. If one want to calculate the symmetries of the theory, it necessary to look for the isometries of the space involved. Newtonian mechanics is invariant under translations and rotations while special relativity is invariant under the whole Poincaré group.

If we change the parametrization of the trajectory of a particle with terms involving acceleration, we will obtain new terms in equations because those reparametrizations are not transformation in the group of isometries of the space considered.

Third: We have, in fact, a theory that solves this problem: general theory of relativity. In this case the spacetime is also a 4d-spacetime manifold with a Lorentzian metric. First step is compute all the isometries of the spacetime. In particular, GR is invariant under diffeomorphisms, i.e. any smooth reparametrization of the world-lines of free particles will give the same equations. In this case those we have chosen to be "free-particles" are particles moving in geodesics in spacetime.

As an example consider a free falling body: Newtonian mechanics says that this body is not a free-particle because gravity acts on it. In the equations of motion will appear a term related to gravity: $$x = x_0+v_0t+\mathbf{\frac{1}{2}gt^2}$$

However in general relativity this term can be absorbed in the parametrization of the time:

$$\nabla_UU=0\Rightarrow \ddot{x}^\mu+\Gamma^\mu_{\alpha\beta}\dot{x}^\alpha\dot{x}^\beta=0$$ where $U$ is the four velocity of the free bo, $\Gamma$ are Christoffel symbols, related with ficticious forces as Coriolis, andthe time is reeascaled $t^\prime\rightarrow t+\frac{1}{2}\frac{g}{v_0}t^2$.

Notice that if a body is not moving freely, that means that it is not moving along a geodesic, and try to reescale the time in order to achive a similar form of the equations of motion will lead to spurious factors, as in special relativity and newtonian mechanics:

$$\nabla_UU\not=0\Rightarrow \ddot{x}^\mu+\Gamma^\mu_{\alpha\beta}\dot{x}^\alpha\dot{x}^\beta=Forces$$