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where ξ α = x 2 α − x 1 α is the separation vector between two geodesics, D / ds (not just d / ds ) is the covariant derivative, and R α βγδ is the Riemann curvature tensor, containing the Christoffel symbols. In other words, the geodesic deviation equation is the equation of motion for masses in curved spacetime ...
Velocity is the speed in combination with the direction of motion of an object. Velocity is a fundamental concept in kinematics, the branch of classical mechanics that describes the motion of bodies. Velocity is a physical vector quantity: both magnitude and direction are needed to define it.
The equation predicts that for short range interactions, the equilibrium velocity distribution will follow a Maxwell–Boltzmann distribution. To the right is a molecular dynamics (MD) simulation in which 900 hard sphere particles are constrained to move in a rectangle.
A critical requirement of the Lorentz transformations is the invariance of the speed of light, a fact used in their derivation, and contained in the transformations themselves. If in F the equation for a pulse of light along the x direction is x = ct, then in F′ the Lorentz transformations give x′ = ct′, and vice versa, for any −c < v < c.
Here, , and will be used to denote the initial velocity, the velocity along the direction of x and the velocity along the direction of y, respectively. The mass of the projectile will be denoted by m , and μ := k / m {\displaystyle \mu :=k/m} .
To derive the equations of special relativity, one must start with two other The laws of physics are invariant under transformations between inertial frames. In other words, the laws of physics will be the same whether you are testing them in a frame 'at rest', or a frame moving with a constant velocity relative to the 'rest' frame.
Unprimed quantities refer to position, velocity and acceleration in one frame F; primed quantities refer to position, velocity and acceleration in another frame F' moving at translational velocity V or angular velocity Ω relative to F. Conversely F moves at velocity (—V or —Ω) relative to F'. The situation is similar for relative ...
where x' is the position as seen by a reference frame that is moving at speed, v, in the "unprimed" (x) reference frame. [ note 3 ] Taking the differential of the first of the two equations above, we have, d x ′ = d x − v d t {\displaystyle dx'=dx-v\,dt} , and what may seem like the obvious [ note 4 ] statement that d t ′ = d t ...