Following is a list of the frequently occurring equations in the theory of special relativity.
To derive the equations of special relativity, one must start with two other In this context, "speed of light" really refers to the speed supremum of information transmission or of the movement of ordinary (nonnegative mass) matter, locally, as in a classical vacuum.
However, light and other massless particles do theoretically travel at
under vacuum conditions and experiment has nonfalsified this notion with fairly high precision.
does act as such a supremum, and that is the assumption which matters for Relativity.
In the following, the relative velocity v between two inertial frames is restricted fully to the x-direction, of a Cartesian coordinate system.
As the relative velocity approaches the speed of light, γ → ∞.
Applying the above postulates, consider the inside of any vehicle (usually exemplified by a train) moving with a velocity v with respect to someone standing on the ground as the vehicle passes.
To visualize this, picture the light being emitted at one point, then having the vehicle move until the light hits the mirror at the top of the vehicle, and then having the train move still more until the light beam returns to the bottom of the vehicle.
This path will help form two-right sided triangles, with the height as one of the sides, and the two straight parts of the path being the respective hypotenuses: Rearranging to get
So, above, the emission and reception of the light both took place in the vehicle's frame, making the time that an observer in the vehicle's frame would measure the proper time.
He then calculates the train's length as follows: However, the observer on the ground, making the same measurement, comes to a different conclusion.
Because the two events - the passing of each end of the train by the post - occurred in the same place in the ground observer's frame, the time this observer measured is the proper time.
As there existed a proper time for time dilation, there exists a proper length for length contraction, which in this case is ℓ.
Also, as length contraction does not affect the perpendicular dimensions of an object, the following remain the same as in the Galilean transformation: Finally, to determine how t and t′ transform, substituting the x↔x′ transformation into its inverse: Plugging in the value for γ: Finally, dividing through by γv: Or more commonly: And the converse can again be gotten by changing the sign of v, and exchanging the unprimed variables for their primed variables, and vice versa.
These transformations together are the Lorentz transformation: The Lorentz transformations also apply to differentials, so: The velocity is dx/dt, so Now substituting: gives the velocity addition (actually below is subtraction, addition is just reversing the signs of Vx, Vy, and Vz around): Also, the velocities in the directions perpendicular to the frame changes are affected, as shown above.
This is due to time dilation, as encapsulated in the dt/dt′ transformation.
This inner product is invariant under the Lorentz transformation, that is, The sign of the metric and the placement of the ct, ct', cdt, and cdt′ time-based terms can vary depending on the author's choice.
Also, sometimes η is replaced with −η, making the spatial terms produce negative contributions to the dot product or spacetime interval, while the time term makes a positive contribution.
These differences can be used in any combination, so long as the choice of standards is followed completely throughout the computations performed.
In general, in special relativity, four-vectors can be transformed from one reference frame to another as follows: In the above,
can be a four-vector representing position, velocity, or momentum, and the same Λ can be used when transforming between the same two frames.
Invariance and unification of physical quantities both arise from four-vectors.
General doppler shift: Doppler shift for emitter and observer moving right towards each other (or directly away): Doppler shift for emitter and observer moving in a direction perpendicular to the line connecting them: If an object emits a beam of light or radiation, the frequency, wavelength, and energy of that light or radiation will look different to a moving observer than to one at rest with respect to the emitter.
If one assumes that the observer is moving with respect to the emitter along the x-axis, then the standard Lorentz transformation of the four-momentum, which includes energy, becomes: Now, if where θ is the angle between px and
, and plugging in the formulas for frequency's relation to momentum and energy: This is the formula for the relativistic doppler shift where the difference in velocity between the emitter and observer is not on the x-axis.
The first is the case where the velocity between the emitter and observer is along the x-axis.
In that case θ = 0, and cos θ = 1, which gives: This is the equation for doppler shift in the case where the velocity between the emitter and observer is along the x-axis.
The second special case is that where the relative velocity is perpendicular to the x-axis, and thus θ = π/2, and cos θ = 0, which gives: This is actually completely analogous to time dilation, as frequency is the reciprocal of time.
So, doppler shift for emitters and observers moving perpendicular to the line connecting them is completely due to the effects of time dilation.