The third law leads to the Principle of Conservation of Momentum. For completeness, we can add to these laws of motion the more general Principle of Conservation of Energy.
If we imagine an object being at rest or uniform velocity in our reference frame, for Newton's laws to apply in other reference frames, they also must move at uniform velocity relative to us. Therefore, any accelerating frame (including a rotating one, since rotation changes the direction of velocity) is not an inertial frame.
Relativity is all about relating events (which have 4 coordinates: x,y,z,t) between inertial frames. For the examples later, we set up two inertial frames with observers O and O'. Think of these frames as train cars, with O stationary and O' moving to the right with velocity u:
For convenience we choose the +x axis to be the direction of motion of the O' frame. Each observer is equipped with an array of synchronized clocks for "observing" (i.e. measuring) event coordinates in their frame. Additionally, we can arrange for the clocks to synchronize when O' passes O, i.e. t'=t=0.
This makes sense to us: lengths are the same between frames, and as for clocks: once synchronized, always synchronized!
The theory is based on two simple postulates:
The second postulate was confirmed by the Michelson-Morley experiment, in concert with others, and leads to verifiable predictions of Einstein's theory (e.g. extended decay lifetimes of fast particles). It is a very powerful one! It says in effect that if observer O' shines a flashlight and observes the speed of the beam leaving it, she will measure c. However, observer O who is watching this from the side of the track will also measure c! The only way this can make sense is if O and O' disagree fundamentally about distances and times. Let's look at some consequences.
Observers O and O have been equipped with the latest model of light clock for measuring time. The light clock has a scale so they can measure time as a light beam travels up the tube. When the beam reaches the top, the clock "ticks" as the beam is reflected back down to the base, where a detector senses it and sends another pulse. The time for the pulse to travel up the tube of length L0 is t=L0/c:
O and O' synchronize their clocks with a single, initial
flash of light when they pass each other at t=t'=0. As the flash of light expands and O' moves, after
a time t it has reached the top of the O tube, but O also measures the flash
has only traveled a height h up the O' tube, where 
. Now since O' must observe the same speed of light and
has no idea that she is moving (inertial frame), she must interpret the
flash reaching this mark as her clock time t'=h/c
. Eliminating h between these relations gives us:
: The Time Dilation Formula
Therefore, O concludes that time is slowed down in O'! But doesn't this just apply to "light clocks"? No, all clocks (heartbeats, decaying atoms, and digital watches) in O' must then run slow. Otherwise, O' would notice the mismatch between these clocks and her light clock, and conclude that she is in motion - not allowed by the Principle of Relativity! Also, by symmetry of the inertial frames, O' must also measure the same time dilation (slowing) of the O clock as it passes by her in the -x direction - weird, but true!
We define the Proper Time as the time between
events as observed at the same location (D
x=0), e.g. for muon decay, this is the
time as measured by a clock traveling along with the muon. The factor 
is known as the Lorentz
factor and occurs frequently in special relativity. Note that g ³ 1 always,
having its minimum value when objects are at rest in our inertial frame.
Also note: If O' is travelling at the speed of light, O never sees the beam climb up the O' clock. i.e. From the "lab" viewpoint, time almost stands still for objects moving close to the speed of light, and near-light-speed observers conclude the same about us!
Now consider attempts by O and O' to measure their relative speed. O' rides on a train car which she has measured to have length L'. Both observers measure the time it takes for the train to pass observer O from the time that the front of train passes O at t'=t=0:
Solving both relations for u and setting them equal gives:
: The Lorentz Contraction Formula.
So we see that the length of an object depends on its motion - this is a real effect!
We define Proper Length L0 of a body, an intrinsic property, as its length when measured at rest - note that this is the maximum length of the body. In the example above, O' rides on the train and so measures its proper length, L'=L0.
Now at some time later, a point at distance x' in the O' frame will be measured by O to have a coordinate 
, so by rearrangement we have the general position formula:
O' drives a car of proper length L0 past O, who has set up sprinkler/sensor units to fire a jet of water at front and rear windows simultaneously. The front sprinkler fires when the car's weight runs over a pressure pad - the rear sprinkler fires when the car's weight comes off a pressure pad.
Due to Lorentz contraction, O knows that the sprinklers should be spaced apart by 
. However, in the O' frame, the car wash is approaching her at velocity -u and is Lorentz-contracted to
. So O' measures that the jet spacing is too small to wash both the front and rear of her vehicle simultaneously!
Yet both O and O' agree afterwards that the windows are clean. As O' sees it, first the front is washed, then there is a delay t' as the Lorentz-contracted car-wash moves rearwards by a distance:
So dividing by its velocity u,
O' measures a time delay
before
the rear sprinkler fires. Therefore, simultaneous events for O (t=0) are not in general simultaneous for O'!
Setting x'=L0 and including time dilation from before we have 
.
.
In general, inertial observers only agree that events are simultaneous when they occur at the same time and the same place. In the case of our window washing device, some observers will see the front fire before the rear, while others e.g. moving on a freeway parallel to the car wash, will observe the rear washer firing first. However, everyone must agree when they compare notes later that both front and rear windows were cleaned, and in general that "cause" preceded "effect".
Putting everything together - carefully! - we can calculate the coordinates of an event in the O' frame given in the O frame:
The inverse transformation for x and ct can be obtained either by a lot of algebra, or simply by switching prime and non-prime coordinates and changing the sign of the relative velocity, u. The result is the same.
[For the mathematically inclined, this transformation can be derived by considering a ray of light moving along the +x axis, whose motion must be described by x=ct and x'=ct' (both O and O' measure the same speed of light). Substitute these into the equation above and its inverse, then solve to find an expression for g ].
We can also show that y'=y and z'=z from symmetry arguments:
Lengths and times between events may differ in different inertial frames, but the right combination of both can be made into an invariant. It can be shown from the transformations that the Spacetime Interval between events, defined as:
gives the same value (in m2) for all inertial observers. This can be used to solve the "Twin Paradox", or just to show that relative to a stationary observer, a moving observer always "ages" less whether they are coming or going!
A moving source "compresses" wave crests ahead of it, and "stretches" them out behind it. Since the time between wave peaks is the just the period T, which remains constant, the wavelength is compressed ahead of the source to 
.
Classical case: In the classical Doppler effect, the period T is just the original wave period, related to the un-compressed source wavelength by 
. This gives us the classical result 
.
Relativistic case: In special relativity, we know that the time period T in the O frame is time-dilated with respect to the proper time in the source (O') frame, i.e.
; here, T0 is the proper time between wave crests=l
0/c. Substituting for this value of T, the observed wavelength is now:
This is for "blueshifted" light; for a receding ("redshifted") source, just change the sign of the velocity u. Note that unlike the classical Doppler shift, which is limited to producing wavelengths 
, there is no such limit to the wavelength shift in relativity. For example, the most distant quasars observed in the universe have "redshifts" 
.
Under a Galilean transformation, differentiating with respect to time t=t''gave us the law of velocity addition: 
.
However, the Lorentz transformation complicates things a little. Writing down expressions for dx', dy', dz' and dt' we find that velocities do not add/subtract as before - this is true even along the y and z axes due to the time-dilation between frames:
One consequence of this is that you cannot "beat the speed of light" by trying to add velocities, e.g. from an inertial frame with u=0.7c, by launching an object with v'=0.6c. (In this case, the resulting velocity measured by O is v=0.986c).
Differentiating the Galilean transformation twice with respect to time, we find that O and O' agree on accelerations, 
and so by Newton's laws they agree on forces and momentum. However, from the Lorentz transformation we find 
, so the same forces would produce different accelerations. Therefore, we need to redefine momentum as something that is conserved in all inertial frames.
In our frame, twins riding train cars approach at velocity +u and -u along the x-axis, and each throws a punch in the ± y direction to try to knock the other sideways. In our frame, from symmetry, the twins' fists have equal and opposite y-momentum and so both stay on their vehicle.
However, riding in the reference frame of twin B for example, B measures the A's punch to be time-dilated, i.e. moving in slow motion by a factor g due to A's approach velocity. But since twin B doesn't knock twin A backwards backwards, he concludes that twin A packs an equal "punch" to his own. Twin B concludes therefore that for A's fist momentum to be equal to his punch, twin A must have a more massive fist, exactly by the same factor g !
So to conserve momentum =mass ´ velocity, we can define a relativistic mass m=g m0 where m0 is the "rest mass" ("proper mass") of the object (in this case, a twin's fist).
Using this correction to the mass, the Relativistic Momentum
: is now conserved in all inertial frames, and Newton's 2nd law still holds if we use:
Furthermore, from the work-energy theorem and Newton's 2nd law, the kinetic energy K of a particle which has been accelerated by a force is now given by:
For small velocities, this formula reduces to the classical
(Binomial expansion).
1. Potential Energy=U, sum of rest masses=
2. Kinetic energy 
3. Heat energy=U, sum of rest masses=
The "relativistic mass" of the system seems to appear (stage 2), then disappear, so is not conserved. Einstein reasoned that a more useful quantity is the relativistic mass-energy, which is conserved, and concluded that "mass" is yet another form of energy, with the conversion from our conventional units (kg) to mass-energy units (J): E=mc2.
In general, the Relativistic Mass-energy of a moving body is given by its (kinetic+rest mass energy), i.e. 
. This new definition of energy is found to be conserved for all inertial frames.
From the new definitions for p and E above, we have the following relations:
(1) 
(2) 
: The invariant rest-mass energy of a system, which is the same number for all inertial observers.
Note that the rest mass M0 of a system of moving particles is not just the sum of their individual rest masses. Add the energies and the momenta first, then combine them to form the rest mass-energy, i.e.
This frame-invariant "rest mass" of the system does have meaning - a box full particles with zero total momentum will have more inertia and a stronger pull from gravity (i.e. weighs more) than a box containing the same particles at rest.
From the new definition of kinetic energy K, the work required to push an object tends to infinity
as và c,
unless m0=0. In this case, we cannot
use our original formulae for E and p, (both give 0/0), but the relations between them still
hold. With m0=0, we have from (2)
above 
. Then from (1) it follows
that v=c, i.e. massless particles (if they exist)
must travel at the speed of light!
In fact, massless particles do exist in Nature - two common examples are the photon and the neutrino. Although they have zero rest mass, it is still possible for them to contribute to the rest mass of a system:
If photons are created in a box with reflecting walls,
such that their total momentum 
,
the box will have an increase in rest mass 
due to the total energy of the photons. This is a real
effect - weighing machines, rocket engines and gravity will register the
larger mass.
Although length, time, and mass are now relative quantities, our inertial observers O and O' must agree on a number of points. Here is an incomplete list:
