Special Relativity

SI units & Physics constants

Special relativity investigates motion of objects at high speeds approaching the speed of light in inertial reference frames

Here (all units see here):

x, y, z is fixed inertial reference frame

x', y', z' is inertial reference frame moving at constant velocity along x-axis

v is velocity of frame x', y', z' with respect to frame x, y, z

m_o is rest mass of the object measured in frame x', y', z'

t is instant of time of position of the object measured in frame x, y, z

t' is proper instant of time of position of the object measured in frame x', y', z'

 is proper force with components F_ox, and F_oy acting on the object measured in frame x', y', z'

c is speed of light

General formulas

Lorentz transformation between coordinates and time for frames x', y', z' and x, y, z

Inverse transformation

Einstein's Postulates of Special Relativity

Relativity Postulate. All laws of physic are the same (have the same form) in every inertial reference frame

Speed of Light Postulate. The speed of light in vacuum, measured in any inertial reference frame, always has the same value, no matter how the source of light and the object are moving relative to each other

Relativistic effects

Time Dilation. The time interval between two events, measured by observer standing in fixed frame x, y, z

where  is proper time interval between the same two events taking place at the same point, measured by observer standing in moving frame x', y', z'

Length Contraction. The length or distance between two points in direction of motion, measured at the same time by observer standing in frame x, y, z

where L_o is proper length or distance between the same points, measured by observer standing in frame x', y', z'

Simultaneity Violation. The time interval between two events which are simultaneous in frame x', y', z', measured by observer standing in frame x, y, z

where L_o is proper distance between the points where the simultaneous events take place, measured by observer standing in frame x', y', z'

Relativistic law of adding velocities. The velocity of object, measured by observer standing in frame x, y, z

where u_o is relative velocity of the object moving  in x-direction, measured by observer standing in frame x', y', z'

Relativistic mass. The mass of object, measured by observer standing in frame x, y, z

where m_o is rest mass, measured by observer standing in frame x', y', z'

Relativistic momentum. The linear momentum of object moving with velocity

Relativistic force vector. The x and y components of force  acting on object, measured by observer standing in frame x, y, z

where F_ox and F_oy are components of proper force, measured by observer standing in frame x', y', z'

The relativistic Newton's Second Law for motion

Total energy of mass moving with speed v

where m_o is rest mass

Rest energy of mass

Relativistic kinetic energy of object moving with speed v

Relation between relativistic energy and momentum