General-Relativity

## Principle of Relativity

The Principle of Relativity states that: The laws of physics take the same form in every intertial frame. No exception has been found to this principle, and it holds equally well in both Newtonian theory and Special Relativity.

## Newtons Laws of Motion

I # In an intertial frame of reference (i.e. a frame of reference undergoing zero acceleration), an object at rest stays at rest (or, similarly, keeps its initial velocity) unless acted upon by an outside force. $$\sum \mathbf{F} = 0 \Leftrightarrow \frac{d\mathbf{v}}{dt} = 0$$ II # In the simplest of terms: $$\mathbf{F} = m\mathbf{a}$$ which states that The vector sum of the forces on an object is equal to that objects mass times the acceleration ...

## Newtonian Spacetime

Newtonian Spacetime differs from Relativistic spacetime in that time is considered an absolute (i.e. $$t^{\prime} = t$$). Transformations between reference frames can be achieved with Galilean transformations. A particle traveling along the x axis in $$S$$ at a constant speed u, has a speed $$u^{\prime}$$ found by: $$u^{\prime}_{x} = \frac{dx^{\prime}}{dt^{\prime}} = \frac{dx^{\prime}}{dt} = \frac{dx}{dt} - v = u_{x} - v\tag{1}$$ remembering that the first derivative of position is velocity, and the second derivative of motion is acceleration ...

## Inertial Frames

Above are two frames in Cartesian coordinates, $$S$$ and $$S^{\prime}$$. We have coordinates $$(x, y, z)$$, which define our dimensions. Inertial Frame # An inertial frame is a frame of reference for which acceleration is zero. In other words: $$\frac{d^{2}x}{dt^{2}} = \frac{d^{2}y}{dt^{2}} = \frac{d^{2}z}{dt^{2}} = 0$$ In the absence of gravity if $$S$$ and $$S^{\prime}$$ are two inertial frames they can only differ from each other by (and/or): ...

## Galilean Transformations

In Newton’s formulation of spacetime, time is an absolute; meaning that every observer experiences the same flow of time. Assuming an inertial frame with only translation across the x axis, an event $$P$$, can be translated between $$S$$ and $$S^{\prime}$$ by the following linear transformations: $$t^{\prime} = t\tag{1}$$ $$x^{\prime} = x - vt\tag{2}$$ $$y^{\prime} = y\tag{3}$$ $$z^{\prime} = z\tag{4}$$ Switching between coordinates ($$S$$ v. ...