Our Place in the Universe

Radar (radio detection and ranging) works by timing the delay between sending a radio pulse and receiving the reflected pulse. By using the speed of light, the total round-trip distance can be calculated, and halved to find the one-way distance. This method uses the relativistic principle of the invariance of the speed of light. Lidar works in a similar way, but using light waves made by lasers instead of radio waves.

NIS: Parallax is never explicitly mentioned in the specification.
Parallax is the shift in an object's position when it is viewed from different locations. This causes nearby objects to appear to shift more (move faster) than distant objects. The baseline is the known distance (either 1 or 2 au). The parallax angle is the apparent angle that the near star has moved through, often measured in arcseconds.

A circle is divided into 360 degrees (360), each of which is divided into 60 arcminutes (60), each of which is divided into 60 arcseconds (60). Thus, an arcsecond represents of a complete rotation.

NIS: Standard candles are never explicitly mentioned in the specification.
Some stars, Cepheid variables, have a variation in brightness that depends on their absolute brightness. The longer the period of the variation in brightness, the brighter the star. By comparing the absolute brightness of the star with the apparent brightness, the distance to the star can be calculated using the inverse square law.

Doppler shifts occur when an observer moves relative to the emitter of a wave. Consider a wave with wavelength , wave speed , and time period , emitted at relative speed to an observer:

In the time between two peaks, the source moves a distance away from the observer:

By considering the increase in wavelength:

The equation holds for non-relativistic speeds (speeds much lower than the speed of light).

The Doppler effect can be used to determine whether distant galaxies are moving towards or away from us. Atoms in distant stars absorb and emit light with specific wavelengths, which produces absorption or emission spectra. The wavelength of these spectra changes according to the Doppler effect:

Hubble found that the further away a galaxy was, the greater the speed at which it was receding:

Where is the recession velocity, is the Hubble constant, and is the distance to the galaxy. has units of (kilometres per second per megaparsec).

Hubble's Law suggests that the Universe expanded from being smaller in the past to its current state. The age of the Universe can be estimated using the Hubble constant:

Cosmological redshift explains the very large redshifts observed in distant galaxies. is the redshift, where . In astronomy, redshifts of up to have been observed; the non-relativistic Doppler equation would imply that these galaxies are moving at , which is clearly not possible.

Cosmological redshift is the redshift caused by the expansion of the Universe. As light travels through the expanding Universe, the wavelength of the light is stretched.

For a wave with wavelength and a distance :

Thus for a red-shift of , space has stretched by a factor of . Thus, when distant objects with are observed, we are looking at when the Universe was around one-eleventh the current size.

The cosmic microwave background is leftover electromagnetic radiation from the Big Bang that has experienced redshift. It was first produced when the Universe was cool enough for atoms to form, emitting photons. These photons initially had a wavelength of around µ, but have since been redshifted to around , giving .

This provides evidence for the hot Big Bang model; it shows that the Universe began in a hot, dense state, and has since expanded and cooled.

Space-time diagrams show objects moving through space and time:

In particle accelerators or cosmic rays, particles travel at a very high speed and experience relativistic effects. The rest energy of a particle is given by:

Where is rest mass, and is the speed of light.

The total energy is given by , where is the relativistic or Lorentz factor, defined as:

A graph of total energy, , against velocity, , shows how as the total energy tends to infinity, the speed approaches . Thus, no objects can reach the speed of light, as this would require infinite energy. A graph of has the same shape.

The following formulae are all given in the formula booklet:

IS: Time dilation is explicitly mentioned in the specification. The derivation, however, is very likely not required.
For an observer, let wristwatch time be the time recorded by a clock travelling with that observer.

Consider a light clock - a pair of mirrors with a pulse of light bouncing back and forth. A thought experiment can be carried out to explain where length contraction and time dilation come from.

For an observer moving with the clock, the light takes to make a round-trip.

For an observer watching the clock travel past at speed , the Pythagorean theorem can be used (where is the time experienced by the clock):

So to the observer, the time taken per tick was , while from the clock's frame of reference, the time taken per tick was . As , the clock ticks 'slower' from the clock's frame of reference.

In general, time passes more slowly for objects moving at high speeds, compared to stationary observers. The time dilation is given by , which is not on the formula booklet.

NIS: Length contraction is not explicitly mentioned on the specification.
Length contraction is another consequence of special relativity.

So to the particle, the length was , while from the stationary observer, the length was . As , the particle travels a smaller distance in its own frame of reference compared to a stationary observer.