universe + special relativity

Question 1

distances and measurements used in universe + special relativity?

Answer 1

distances and measurements used in universe + special relativity are:

• lightyear: the distance light travels in one year (1 ly = 9.5x10^15m)
• astronomical unit: the average distance between the earth and sun (1 AU = 1.5x10^11m)
• arcsecond: 1/3600 th of a degree
• arcminute: 1/60 th of a degree
• parsec: the distance away an object must be to have a parallax of one arcsecond when observed from Earth (1 pc = 3.1x10^6m or 1 pc = 3.3ly)

Question 2

explaining RADAR

Answer 2

• RADAR stands for Radio Detection and Ranging
• RADAR makes use of radio waves to determine the distance to an object, hence speed and acceleration can be determined by calculation
• RADAR only works for short distances because the time delay (Δt) becomes greater and the signal becomes weaker for larger distances

Question 3

how to calculate the distance to and velocity of an asteroid?

Answer 3

1) send out a pulse, pulse then returns, record the time taken for pulse to return (Δt1)
2) wait a certain amount of time (eg 100 seconds)
3) send a second pulse, pulse then returns, record time for pulse to return (Δt2)
4) find the distance the asteroid is at before ‘100s’ by multiplying each each recorded time (from 1 and 3) by the speed of light (the speed of radiowaves)
5) find Δs (the distance travelled in the certain time)
6) relative velocity = Δs / (certain time)

use relative velocity =
Δs / (t2 - t1)

Question 4

what are the assumptions when determining asteroid speed?

Answer 4

some of the assumptions when determining asteroid speed are:

1) speed of the signal is the same both ways (constant speed of light)
2) moment of reflection is halfway through each time delay (Δt)

Question 5

what is the doppler shift? and what can it be used for?

Answer 5

• the doppler shift is a change in the observed wavelength due to the relative motion of the source and observer
• spectral lines are caused by atoms absorbing / emitting light at a particular wavelength…
•… receding stars show red shift, approaching stars show blue shift

Question 6

what is the doppler effect? what happens when source moves towards observer + what happens when source moves away?

Answer 6

• the doppler effect is the apparent change in the observed frequency (and wavelength) of a wave when the source is moving relative to the observer
• if the source is moving towards the observer, the frequency appears to increase and the wavelength appears to shorten
• if the source is moving away from the observer, the frequency appears to decrease and the wavelength appears to lengthen

Question 7

what is a parallax?

Answer 7

⋅ a parallax is the apparent change in position of an object (eg close stars) relative to a fixed background when viewed from a changing angle

Question 8

what is the parallax angle?

Answer 8

⋅ the parallax angle (θ) is half the angle that the object (eg close star) appears to move through (in 6 months)

Question 9

what happens when θ is really small? (parallax)

Answer 9

⋅ when θ is really small (concerning parallax), tanθ (in degrees) = θ (in radians)

Question 10

what is the equation for parallax?

Answer 10

⋅ the equation for parallax is:

(in degrees:) tanθ = 1 AU / d
(in radian:) θ = 1 AU / d

where:
⋅ AU is astronomical unit
⋅ θ is parallax angle
⋅ d is distance from earth to object (eg close star)

Question 11

what happens if d decreases? (parallax equation)

Answer 11

⋅ when d decreases (in the parallax equation), the more parallax it shows as the earth orbits the sun
• bc parallax angle ∝ 1/d

Question 12

what is a standard candle?

Answer 12

⋅ a standard candle is a star of known luminosity that is used as a comparative measure for other stars

Question 13

what is the doppler shift?

Answer 13

⋅ the doppler shift is a change in the observed wavelength of a wave due to the relative motion of the source and the observer

Question 14

what can the doppler shift be used for? (not cosredshift)

Answer 14

⋅ the doppler shift can be used to tell if stars are receding or approaching us:
1) spectral lines of a star are caused by atoms absorbing / emitting light at a particular wavelength
2) receding stars show red shift and approaching stars show blue shift

Question 15

what is the equation used for doppler shift?

Answer 15

⋅ the equation used for doppler shift is:

z = Δλ / λ = v / c

where:
⋅ z = fractional increase in λ (wavelength)
⋅ λ = emitted wavelength
⋅ Δλ = change in wavelength
⋅ v = recessional velocity (of star)
⋅ c = speed of light (can also just be speed of wave)

NOTE:
⋅ equation is correct for vs much smaller than speed of light- for vs near speed of light, special relativity modifies this

Question 16

what is cosmological redshift?

Answer 16

⋅ cosmological redshift is the stretching of the wavelength of light to longer wavelengths as the space it is travelling through expands

Question 17

what is hubble’s law?

Answer 17

⋅ hubble’s law states that “distant galaxies are moving away at a speed proportional to their distance”

Question 18

what is the equation for recession velocity? (cosmological redshift)

Answer 18

⋅ the equation for recession velocity ( in cosmological redshift) is:

v = (H0)r

where:
⋅ v = recession velocity
⋅ H0 = the Hubble constant
⋅ r = distance

Question 19

what is Hubble time?

Answer 19

⋅ Hubble time is the time since galaxies were close together (as a singularity) i.e. the time scale / age of the universe
⋅ it is calculate using:

1 / H0 = r / v

⋅ the bigger H0 means the younger the universe is, so the faster the universe must have expanded to reach its present size

Question 20

what is the big bang?

Answer 20

⋅ the big bang is the probable origin of the universe - a very hot, dense state from which it has expanded and cooled

Question 21

what is cosmic microwave background radiation?

Answer 21

⋅ cosmic microwave background radiation (CMBR) is microwave radiation that permeates the universe
⋅ CMBR is red shifted radiation from an early hot state of the universe
⋅ CMBR has the largest known redshift, having been produced with a wavelength of the order 1μm to now the order 1mm (1000x larger)

Question 22

what has happened to the temperature of CMBR as the universe has redshifted until now?

Answer 22

⋅ the temperature of CMBR has fallen by a factor of 1000 (from 3000K to 3K) (inverse of wavelength of CMBR, which has increased by a factor of 1000) as the universe has redshifted until now

Question 23

is there variation in the temperature of CMBR across the universe? and what is believed to have formed from these variations?

Answer 23

⋅ yes there are variations in the temperature of CMBR across the universe
⋅ it is believed that stars and galaxies formed from these non-uniformities
⋅ the cosmological expansion has stayed the same however

Question 24

what is special relativity?

Answer 24

⋅ special relativity explains the motion of bodies with very high speeds, speeds close to the speed of light
⋅ (low speeds produce the results as predicted by Newton’s Laws)

Question 25

what was Einstein’s First Postulate?

Answer 25

⋅ Einstein’s First Postulate was that: “the laws of physics are the same for all observers in all inertial frames of reference”.
⋅ so physical behaviour cannot depend on any absolute velocity

Question 26

what was Einstein’s Second Postulate?

Answer 26

⋅ Einstein’s Second Postulate was that: “the speed of light is the same for inertial frames of reference”.
⋅ the speed of light does not depend on the motion of the light source or observer (in other words, the speed of light in free space is invariant)

Question 27

what is the Lorentz Factor?

Answer 27

⋅ the Lorentz Factor (relativistic factor) (γ) is the factor by which time changes for an object, while that object is in motion as measured by an observer
⋅ it can be calculated using the equation below:

where:
⋅ γ = lorentz factor
⋅ v = speed of object
⋅ c = speed of light

NOTE:
⋅ if v tends to 0, γ tends to 1, so t = t0, same as predicted by Newton’s Laws
⋅ if v = c, we get γ = 1/0 which is undefined, so this supports the idea that nothing can go faster than the speed of light (superluminal travel is impossible)

Question 28

what are inertial frames of reference?

Answer 28

⋅ inertial frames of reference are frames with a constant or zero velocity (i.e. where Newton’s Laws apply, as there is no net external force acting on the frame)
⋅(nothing is truly ‘at rest’ or truly ‘moving’, only relative to another object)

Question 29

what are space-time diagrams?

Answer 29

⋅ in space-time diagrams, the line that traces out on a space-time diagram is called a worldline
⋅ for an object moving at non-zero relative velocity, its worldline moves up the time axis (y-axis) to or away from the observer (x=0)
⋅ for an object with zero relative velocity, it traces a vertical worldline on the space-time diagram

⋅ a light pulse travels at 1 light second per second so has a worldline at 45 degrees
⋅ due to the second postulate, this is true for every space-time diagram
⋅ a steeper gradient means the object is moving slower (moving less distance during a longer time)
⋅ a worldline cannot have an angle less than 45 degrees as objects cannot travel faster than the speed of light

Question 30

what are some of the assumptions for space-time diagrams?

Answer 30

in space-time diagrams, we assume:
⋅ that the speed of light is constant, so the reflection occurs halfway through the ‘flight time’ of the pulse to and from the object
⋅ that the speed of light is not affected by the motion of the distant object

Question 31

what does time dilation mean?

Answer 31

⋅ time dilation means that the amount of time between two events differs depending on whether you are at rest relative to them
⋅ eg) clocks moving relative to an observer run slowly as seen by the observer

Question 32

describe the Time Dilation and Light Clock Thought Experiment?

Answer 32

in the light clock thought experiment:
1) a pair of mirrors between which light bounces
2) one tick is when the light travels to the other mirror and back
3) consider a stationary light clock on a train where the light pulse bounces vertically, ie as viewed by a person on the train
4) the distance to the other mirror is d = (c x t0), so the time to return to the same mirror is 2t0; t0 is the time as measured by a stopwatch on the train
5) the train moves with a speed (v) so an observer outside the train sees the train measured by a stopwatch in the frame of the observer
6) according to the observer, the light pulse moves at an angle a distance (c x t) there and a distance (c x t) back to the first mirror
7) t > t0 so ct is greater than ct0, hence the observer sees the clock ‘tick’ slower when the clock is moving
8) using a diagram, this mathematical derivation can be seen to only need Pythagoras’ Theorem:

Question 33

what is the equation for ‘dilated time’?

Answer 33

the equation for dilated time is:

t = γ x t0

where:
⋅ γ is the lorentz factor