ODU Resources

Updated for the 2018 changes

Part 1, containing notes, tutorials and practicals



Part 2 of the notes in word format, you can adapt these if you can open them.

These are part 2 of the notes in pdf format, so you all ought to be able to open them.


Well after spending 18 months or more several years ago putting everything together students have unanimously declared they want everything separated, so your wish is my command students- here is the complete Our Dynamic Universe section notes with nothing but the essential practicals plus one!

These are part 1 of the notes in pdf format, so you all ought to be able to open them. There is a word version underneath.

These are part 1 of the notes in word format, you can adapt these if you can open them.


Teamwork by Mr Stewart (Berwickshire HS) and I. He designed and made them and I tweaked them. Thanks Mr Stewart they’re ace!

Thanks Mr R Stewart- what a team!

Thanks Mr R Stewart!

For those having trouble with Unit 1 part 1 try this little document

1. 1a Equations of motion

1. 1a Equations of motion

I’ve removed the Time Dilation detailed version and added it as a separate document as I suspect most of you wont read them; which is a pity as it makes everything seem fine! Based on Russell Stannard’s excellent book “Relativity- a very short introduction” Oxford. (2008)  ISBN 978–0–19–923622–0)

ODU worked ANSWERS_4 Currently the most up to date version of the worked answers.

ODU worked ANSWERS_4 The pdf version of the most up to date version of the worked answers.

Additional Support

Chapter 1 exam questions B for CFE higher

Chapter 1 exam Answers B for CFE higher

These are powerpoints prepared for the Revised Higher in 2000. They are still relevant now, and talk through example questions. They are great for revision.

It might be old, but sometimes the old ones are the best. Link for the ppp below!

Linked to some talking questions and answer. ppp below

For those struggling with the vectors try these to give you some practice Great Resource from Mr Crookes. Set up your 2 vectors, either use a scale diagram or components and compare to the given answer. Enjoy!

If you don’t like proving v2=u2+2as from v=u+at then use this neat little sheet from Mr Mackenzie.

A lovely little summary from G Gibb!

Equations of Motion

4.4 ODU EqoM 2012 this document has the macros enabled (actually I think you might need to contact me to get the macros, they are not allowed to be uploaded on a WordPress Website. It allows you to check your answers for the acceleration time graphs that you drew from the velocity time graph diagrams.

using displacement equation to prove the last equation

Click on the image to open a power point of Adding Vectors.

Forces, Energy and Power


africanfastfood This is an introduction to the momentum topic; think about the collision and where the energy is transferred.

Collisions- Think Safety before buying a car!


Projectiles thanks to Mr. Rossi for this one.

Battleships & AWACS Projectiles thanks to Mr. Rossi for this one too.

Special Relativity

Time dilation02

Cleonis [CC BY-SA 3.0 (http://creativecommons.org/licenses/by-sa/3.0/)]

The green dots and red dots in the animation represent spaceships. The ships of the green fleet have no velocity relative to each other, so for the clocks onboard of the individual ships, the same amount of time elapses relative to each other, and they can set up a procedure to maintain a synchronized standard fleet time. The ships of the “red fleet” are moving with a velocity of 0.866 of the speed of light with respect to the green fleet.

The blue dots represent pulses of light. One cycle of light-pulses between two green ships takes two seconds of “green time”, one second for each leg.

{\displaystyle {\sqrt {3}}}

As seen from the perspective of the reds, the transit time of the light pulses they exchange among each other is one second of “red time” for each leg. As seen from the perspective of the greens, the red ships’ cycle of exchanging light pulses travels a diagonal path that is two light-seconds long. (As seen from the green perspective the reds travel 1.73 ({\displaystyle {\sqrt {3}}}) light-seconds of distance for every two seconds of green time.)The animation cycles between the green perspective and the red perspective, to emphasize the symmetry.

OnVelocities This is a document referred to in the Research Task in the ODU part 2 notes.

PHYSICS WORLD ARTICLE DECEMBER 2009 This is a document referred to in the Research Task in ODU part 2 notes

The Expanding Universe

The expanding universe powerpoints. Might not be quite the final version

This is the pdf version of the powerpoint

The above is the pdf version of the powerpoint

Are we missing something in the Expanding Universe?

AH (Doppler)– some of this is relevant to Higher.


The homework booklets are now in the HOMEWORK section.

Homework Booklet Complete pp6-8 (first question), 10-16, 18. Complete notes on Units prefixes and Sci Notation, Uncertainties, Equations of Motion. Read up on Forces.

Updated August 2019

Jacqui Lawson Cards does Physics

 Hickory Dickory Dock

I received this card several years ago (thank you Mr and Mrs Long), and it shows Physics in the animated world!

My immediate reaction to the card was “What big ceilings you have Chudleigh.”

With a stopwatch and the assumption that g remains constant Higher and A level students should calculate the minimum height of the room Chudleigh finds himself in. With love and best wishes Mrs Physics.
Make sure you add your answers in the comment box below! Working optional

Hickory Dickory Dock

Hickory Dickory Dock Answers pdf

Hickory Dickory Dock Answers word

OK, here goes, my results

Time for cork to go up and return on average 6.6 s.

That gives a time to reach the max height 3.3 s

At height v=0, a=-9.8 ms-2, t=3.3s.

Using s=ut+1/2 at2 we can estimate the maximum height the cork reaches

Now the best way to work out s is putting this result into the equation for height

Now I don’t advertise on this site, but if Jacquielawson.com allows me to keep this up then I assure you I am happy to link and tell you that this site contains the best e-cards you can want and for the unlimited number you can send it is worth every penny. They really do cheer up people



What is the Biggest Ever Redshift?

A discussion on the Physics Teachers’ Network requested advice on “What is the biggest ever redshift detected?”

Research shows it was a redshift, z = 11.09 for galaxy GN-z11; and the measurements were  taken in the near infra red using Hubble’s Wide Field Camera. 

This is a big question because effectively we are seeing the furthest galaxy back in time.  It is 32.2 billion years away and came into existence 400 million years after big bang.  So if the Universe is only 13.8 billion years old then how come we can see something so far away?

During this time the Universe was opaque and full of neutral atoms.

Professor Martin Hendry supplied an interesting reply.

In some cases we can determine the redshift of a galaxy by measuring the wavelength of a particular spectral line that corresponds to a particular transition of an electron in a hydrogen atom.  For example the Lyman alpha emission line is the result of an electron dropping down from the n=2 energy level to the n=1 energy level, and the presence of this spectral line is often seen as an indicator of a recent burst of new stars forming as one might expect to see in a very young, recently formed galaxy.  (This line was proposed as a tell-tale sign of a very young galaxy by Bruce Partridge and Jim Peebles – awarded the Nobel Prize for physics this week: see e.g. https://en.wikipedia.org/wiki/Lyman-alpha_emitter).  This line has a wavelength of 121.567 nm in the rest frame of the hydrogen atom.  If a galaxy is a strong Lyman alpha emitter, and the line is observed at wavelength lambda, then by comparing the observed wavelength with the 121nm at which it was emitted we can measure the redshift of the galaxy.

(Of course if this spectral line is redshifted then how do you know it’s a Lyman alpha line?  Likewise for any other spectral line.  Often it’s the combination of several spectral lines and their relative spacing that gives the game away – a bit like a bar code in the supermarket.  You could imagine enlarging the image of a bar code in a photocopied and, generally, it’d still be recognisable as the overall pattern would still be the giveaway).

In fact for this record-holding galaxy, the redshift was determined a slightly different way, from the Lyman series but not the Lyman alpha line and not from an emission line but an *absorption* line: specifically it was determined from the “Lyman break” – i.e. the limiting wavelength that corresponds to the amount of photon energy you need to absorb to allow an electron in the n=1 energy level to escape from its hydrogen atom altogether.   That is a higher energy (and so a higher frequency, and a shorter wavelength) than the Lyman alpha line, and in fact corresponds to about 91 nm in the rest frame of the hydrogen atom.   Any photons that have even higher energies (and thus even shorter wavelengths) than this get absorbed by the (lots of) neutral hydrogen that is around in the Universe at that time; these photons thus *ionise* that neutral hydrogen.  This is sometimes referred to as “re-ionisation” in the sense that the universe was fully ionised when it was much younger, because it was much hotter, then it cools enough for neutral hydrogen to form – i.e. when the CMBR was emitted – and now it’s being ionised again.  Where are the high-energy photons coming from to do this ionising (being absorbed in the process)?  They are believed to come from hot young stars – i.e. the newly formed stars in these young galaxies.  (Remember, the more massive the star the hotter their surface temperature, so massive blue stars emit lots more of these energetic photons than cooler red stars do).

So, in summary, the spectrum of light from a galaxy as a whole drops off at the Lyman break, like a “cliff edge” because at shorter wavelengths than the Lyman break these photons get absorbed, ionising the hydrogen gas in their environments.

You can then play the same game as with an emission line: look for where this “cliff edge” appears in the observed spectrum and then use that observed wavelength (which will be much longer than 91nm) to estimate the redshift.

The research paper on GN-z11 is at https://arxiv.org/pdf/1603.00461.pdf, and is actually pretty readable I think…

Other references:



Another clear explanation from Prof. Hendry, who never makes us teachers feel silly for asking questions. Thanks to Mr Thomson and his student for the original question.

The Expanding Universe

The Fate of the Universe



Here are some great little videos.

The Expanding Universe-NOVA

Here are some more fantastic videos, which you generally can’t have access to in school so I’ve downloaded the videos for you. When at home please visit the original sources.

Dr Physics A is a fantastic communicator who makes all the content so accessible. He is working through the A level course and some of the content relates to A level material, and is done in a different manner to Higher and AH. The content isn’t wrong, it is just defined by different terms. Find his channel here.

Dr Physics A

Watch, if you check his equations for the special relativity topic he transposes t and t’ (as do many texts etc. Scotland always does things differently)! Check off each of the content points from the outcome booklet.


Hubble’s Constant The Big Bang Theory- Part 1

Hubble’s Constant The Big Bang Theory- Part 2

Hubble’s Constant The Big Bang Theory- Part 3

From the Big Bang to Now

Olber’s Paradox

Dark Energy

BBC Dark Energy


Special Relativity

Resources for Special Relativity

Here is a link to a fantastic little book that started me on my “very short introduction” library. It has been uploaded as a pdf file, but if you enjoy it give the author some credit and pay the guy (Russell Stannard) by buying it!


Frames of Reference

You should all try to make your holiday videos so useful in showing Physics ideas! Who is in motion? Does it remain the same throughout the sequence?

This is covered in the web-based research post but I’ve uploaded it here as an MP4 file.

Just check this off against the content as it isn’t all covered at Higher (some is the AH and some isn’t covered at all).

Neil deGrasse Tyson with his inimitable style explains the Michelson-Morley experiment and shows that despite getting a rubbish result it doesn’t say your results are rubbish! This was big Science progress and it wasn’t explained until Einstein came along. It was the turning point that transformed Science.

Here are further explanations of the Michelson-Morley experiment and a hint of more of the course to come.

Evidence for Special Relativity

Sixty symbols- Nottingham University

Sixty Symbols by Nottingham University are an amazing set of videos, although far more than sixty by now. Check out and keep watching.

…. and here at the end I have uploaded the worked answers (thanks to whoever wrote these excellent questions) so that you can check off your tutorials.

ODU worked ANSWERS_5

Our Universe tutorial solutions


updated October 2019

Perfect Answers (according to the SQA)

I will add and adapt these as I go through the past papers. If you’ve got additional definitions do pass them on. These would be great written on cue or flash cards. Go through past papers and record any common phrases or answers.

perfect answers.xlsx

perfect answers.xlsx
1accelerationAcceleration is the rate of change of velocity
2acceleration of  X  ms-2The velocity of the body increases/changes by  X  m s-1 every second.
3Air bags / crumple zones / seat beltsTime (of collision) increased, change in momentum is the same, (Average) force (acting on passenger) is decreased/reduced/smaller
4Big Bang DescriptionThe Universe was initially in a hot and very dense state and then rapidly expanded. The universe started from a point/singularity and rapidly expanded.
5Big Bang EvidenceCosmic Microwave Background Radiation, Redshift of galaxies, Olber’s Paradox (and darkness of the skies), H-He Problem (relative abundance of hydrogen/helium)
6Closed universeThe universe will slow its expansion and eventually begin to contract.
7Collisions- law of conservation of momentumDuring any collision, in the absence of external forces momentum is conserved, total energy is conserved. In an elastic collision – Ek is also conserved In an inelastic collision- Ek is transferred into other forms.
8Component of weight down a slopeW sin θ, or mg sin θ
9Cosmic Microwave Background Radiation. CMB Give a reason why the existence of this radiation supports the Big Bang Theory.is pervasive throughout space. It is the dominant source of radiation in the Universe. It is very uniform (throughout the Universe). It is isotropic (throughout the Universe). It shows the characteristics of blackbody radiation. It has a temperature of approx 3 K (2·74 K) due to cooling on expansion. It corresponds to a redshift of 1000, so the early temperature of this radiation was approx 3000 K. CMBR is thought to be the “afterglow” of the Big Bang, cooled to a faint whisper in the microwave region.
10Dark Energya hypothetical form of energy whose negative pressure counteracts gravity and is assumed to be responsible for the universe expanding at an accelerating rate. Dark Energy repels.
11Dark Mattera hypothetical form of matter invisible to electromagnetic radiation, postulated (suggested) to account for gravitational forces observed in the universe. Dark Matter is attractive
12Doppler EffectThe Doppler Effect is the apparent change in frequency of a wave when the source and observer are moving relative to each other.
13Doppler explanationThere are more wavefronts per second observed due to the forward motion of the vehicle. (NB not wavefronts produced as there are not!)
14Evidence for the big bangThe cooling of the Universe and cosmic microwave background radiation provide evidence for the Big Bang
15Explain why a greater number of muons are detected on the surface of the Earth than would be expected if relativistic effects were not taken into account.For an observer on Earth’s frame of reference the mean life of the muon is much greater OR The distance in the muon frame of reference is shorter
16Explain why star is redshifted.The star is moving away from the Earth • The apparent wavelength of the hydrogen spectra from the star has increased • The apparent frequency of the hydrogen spectra from the star is less than the actual frequency on Earth • The frequency of the light from the star has shifted towards the red end of the spectrum • Light from the star is experiencing a Doppler shift.
17Explain why the driving force must be increased with time to maintain a constant acceleration.The faster it goes, the greater the air resistance. or frictional forces / friction / drag then F(drive) constant, the unbalanced force would decrease or increasing F(drive) keeps the unbalanced force constant or overall/net force – must have
18Explain, in terms of the forces, why there is a maximum angle of slope that the cart can ascend.As angle (of slope) increases mgsinθ increases When mgsinθ ≧ engine force – friction, the vehicle cannot move up the slope
19F-t graphsThe Area under a F-t graph is equal to impulse or change in momentum of the object.
20Hubble Constant Ho and 1/Hothe ratio of the speed of recession of a galaxy (due to the expansion of the universe) to its distance from the observer. The reciprocal of the constant is called Hubble time and represents the length of time for which the universe has been expanding, and hence the age of the universe.
21impulseThis quantity Ft is called the IMPULSE and it is equal to the CHANGE IN MOMENTUM of the object. Impulse = change in momentum
22inertiaInertia is the tendency of a body to remain at rest, or if moving, to continue its motion in a straight line
23inertial reference frametwo objects that are moving at constant speed with reference to each other
24Law of conservation of momentumTotal momentum before (a collision) is equal to the total momentum after (a collision) in the absence of external forces
25Length contractionThe decrease in length (in the direction of motion) of an object moving relative to an observer. N.B. it must be clear that the observer is in a different frame of reference.
26Lorenz transformation not noted at low speeds.Lorentz factor is (approximately) unity/equal to one negligible change in length/time/mass observed
27momentumthe product of mass times velocity
28Newton- definitionOne Newton is equal to the force which causes an acceleration of one metre per second squared when applied to a mass of one kilogram.
29Open UniverseThe universe will continue to expand forever.
30Radiation of peak wavelength 1·06 mm can be detected on Earth coming from all directions in space.Cosmic Microwave Background All three words required (Radiation) “CMBR” – Not acceptable, as this is not “naming”.
31Redshift exampleMore distant galaxies are moving away at a greater velocity/ have a greater recessional velocity
32Redshift explanationLight from objects moving away is shifted to larger wavelength or the rate of change of wavelength/emitted wavelength as the galaxy moves away
33Redshift, zRedshift, z, of a galaxy is defined as the change in wavelength divided by the original wavelength, and given the symbol z. It has no units.
34Reference FrameThe background frame against which measurements are made. There is no absolute reference frame.
35Resultant forceA single force that has the same effect as the forces actually acting on an object
36Satellites a curved pathConstant horizontal speed but are accelerating in the vertical direction under the influence of a gravitational field.
37scalar and vectorA vector is a quantity which has both a magnitude and direction. A scalar is fully described by its magnitude.
38Spectral linesA spectral line is a dark or bright line in an otherwise uniform and continuous spectrum, resulting from emission or absorption of light in a narrow frequency range. Spectral lines are often used to identify atoms and molecules from their characteristic spectral lines. These “fingerprints” can be compared to the previously collected “fingerprints” of atoms and molecules, and are thus used to identify the atomic and molecular components of stars and planets.
39Terminal VelocityA constant velocity of an object when the driving force acting on an object is balanced by the frictional force.
40Time dilationTime dilation is a difference in a time interval as measured by a stationary observer and a moving observer.
41twin paradoxSpecial Relativity would suggest one twin who has been out in space or travelling at high speed relative to one remaining on Earth should come back younger, but to the travelling twin they could consider themselves stationary as the Earth moved away fast, so to the astronaut twin the twin on Earth should be younger. This paradox is resolved as during the journey in space the travelling twin would have had to return, during which time they would be in a non-inertial reference frame, so they would have experienced the speed/acceleration. Therefore been in a non inertial reference frame and hence special relativity does not apply.
42Speed of light is the same for all observers / all (inertial) frames of reference
43Correct – from the perspective of the stationary observer there will be time dilation Incorrect – from the perspective of the students they are in the same frame of reference as the clock Not possible to say/could be both correct and incorrect – frame of reference has not been defined
44light is redshifted/ shifted towards red

Proving the equations of motion

Equation 1

v u at

Equation 2

v-t graphs ut 12at2


Equation 3

EOM equation 3

If you don’t like proving v2=u2+2as from v=u+at then use this neat little sheet from Mr Mackenzie.

using displacement equation to prove the last equation

Below are the 16 graphs to try to learn. Going across a row from L to R shows graphs drawn from the gradient of the previous graph. Therefore if you write the axes on for s-t, v-t and a-t you can work backwards and forwards to quickly picture the shape of the graph you require.16graphs

Friction always acts to oppose motion. If the object is sliding down the slope then friction must act up the slope, but if the object is being pushed up the slope then friction acts down the slope.friction down slopefriction up slopeThe component pushing into the slope (mg cosθ) is balanced by the reaction force from the slope.



Collisions Seatbelts, crumple zones, airbags, helmets etc are all designed to reduce the force on a person, by increasing the time of contact. In all cases the change in momentum or impulse remains the same as the vehicle/ object still has to come to rest from its initial speed.lower force


finding g

Show that the frequency f of the sound heard by the passenger is given by where symbols have their usual meaning.Doppler proof

Although we talk about the Big Bang, it is important to emphasise that the universe, ie space, is expanding. There are a number of characteristics that indicate it is an expansion and not the result of an explosion.

Explosion Expansion
Different bits fly off at different speeds Expansion explains the large-scale symmetry we see in the distribution of galaxies
Fast parts overtake slow parts Expanding space explains the redshifts and the Hubble law
Difficult to imagine a suitable mechanism to produce the range of velocities from 100 kms–1 to almost the speed of light Expansion also explains redshifts and the Hubble law even if we are not at the centre of the universe


Type of Collision Momentum Kinetic Energy Total Energy
Elastic Conserved Conserved Conserved
Inelastic Conserved Reduced Conserved
Explosions Conserved

zero at start and finish

here Ep is converted to Ek so Ek increases Conserved

Hope these are helpful. Let me know if you want me to add anything further.


Dark Matter v Dark Energy

Here are some links for a more in-depth experience of Dark Matter and Dark Energy. (remember Energy ⇒Expansion, Matter ⇒sMaller, or MAtter⇒ Attracts)


34 mins interesting video.


11 mins on Dark Matter

Brand New Research- could you be a part of this?





The Expanding Universe Practical

Try the following practical


expanding universe school observatory


Expanding Universe Experiment

To understand how the redshift of galaxies is due to the expansion of the Universe, try the following experiment.

You will need the following items:

  1. A round balloon (do not use a long, thin one).
  2. Some coloured stick-on dots (at least 5 different colours).
  3. A piece of string about 50cm long.
  4. A ruler.
  5. A stopwatch or other timer.

Step 1 : Setting Up

You will need to work in teams of at least two, one to blow up and hold the balloon and the other to make the measurements.

Before you start, draw a table for your results like the one below with the colours of your five dots in the 1st column:

Colour of Dot First Distance D1 in cm Second Distance D2 in cm Change in Distance        D2 – D1 in cm Speed v in cm/second
Time to fully inflate the balloon:        seconds.

Step 2 : Making the Measurements

Putting dots on the small balloon

Blow up the balloon a little bit and hold the “nozzle” closed, but do not tie it up.

Stick your five dots onto the balloon. Try to spread them out over the whole balloon.

stick dots

Each of the dots represents a whole galaxy, with the surface of the balloon being the Universe that they exist in.

Choose one of the dots to be your “home”. You can choose any of them.

Step 3

Use string to measure the distance between two dots

measure dots

While one of you holds the balloon, the other one can use the string to measure the distance from your “home” dot to one of the other dots.

Now measure the string distance with a ruler.

When you have measured the distance, write it down in your table in the D1 column.

Step 4

Measure the distances from the “home” dot to all the other dots as well and fill in that column of the table.

Note: The distance from your “home” dot to itself is zero.

Step 5

Now carefully blow the balloon right up, using the stopwatch to time how long it takes. Write down the time in seconds.

Step 6

Now re-measure all the distances from “home” to all the other dots and write then down in the D2 column of your table. Don’t forget that the distance from your “home” dot to itself is zero.

You now need to work out the speed of each galaxy. Remember that:

Here, the Distance travelled is the difference between D1 and D2, so calculate D2 – D1 for each of our dots and write them in the 4th column on the table.

Step 7

The Time taken is the time to blow the balloon up. Work out the speed V for each dot and put it into the 5th column. Because your “home” dot has not moved, its speed will be zero.

Step 8

We are studying how the speed that galaxies seem to have gets larger for galaxies that are further away.

The best way to see this is to plot a graph showing the distance along the bottom axis and with the speed up the side.

This means that you need to plot a graph with axes like the one below:

speed v distance

Put the points for all your dots on the graph using D2 as the Distance.

Step 9 : What does it all mean ?

Use the ruler to draw a straight line that goes as close to as many of the points as possible (don’t forget the “home” dot!)

Think about the following questions and discuss them:

  • Are the speeds of all the dots the same?
  • If not, do they get faster or slower as they get further from the “home”?
  • What would be different if you had chosen a different “home”?
  • What would have been the same?
  • What do you think this tells you about the way that the Universe expands and the redshift of galaxies?

If you are not sure about some of the questions, can you think of a way changing the experiment to make them easier to answer?




The Twin Paradox

Citation for this article:
Markus Pössel, ” The case of the travelling twins ” in: Einstein Online Vol. 04(2010), 1007

The case of the travelling twins- let’s call them Pat and Mike!- pictures to come!

An article by  Markus Pössel

In Einstein’s special theory of relativity, there is no such thing as “time” in the singular. Time passes differently for different observers, depending on the observers’ motion. The prime example is that of the two hypothetical twins: One of them stays at home, on Earth. The other journeys into space in an ultra-fast rocket, nearly as fast as the speed of light, before returning home:


Afterwards, when the twins are reunited on Earth, the travelling twin is markedly younger, compared to her stay-at-home sibling. The exact age difference depends on the details of the journey. For example, it could be that, aboard the space-ship, two years of flight-time have passed – on-board clocks and calendars show that two years have elapsed, and both spaceship and travelling twin have aged by exactly that amount of time. On Earth, however, a whopping 30 years have passed between the spaceship’s departure and its return. Just like all other humans on the planet, the twin on Earth has aged by 30 years during that time. Seeing the two (ex?) twins side by side, the difference is striking.

So far, so strange, but undoubtedly real. Space-travel with speeds close to that of light may be unfathomably far beyond the reach of current technology. But sending elementary particles on round trips in a particle accelerator at 99.99999 percent of light speed is routine. The result is in precise agreement with the predictions of special relativity – the “inner clock” of such a travelling particle runs much slower than that of a particle of the same species that remains at rest.

Turning the tables?

The reason the case of the travelling twins is also known as the “twin problem” or even the “twin paradox” is the following. From the point of view of the twin on Earth, one can explain the age difference by appealing to time dilation a basic concept of special relativity. It involves an observer (more precisely: an inertial observer), for instance an observer that lives on a space station floating through empty space. For such an observer, special relativity predicts the following: For any moving clock, that observer will come to the conclusion that it is running slower than his own. Whether it is a clock on another space station floating past or a clock on an engine-driven rocket, in the time it takes for a second to elapse on the observer’s own clocks, less than a second will have elapsed on the moving clock. This slowdown is true not only for clocks, but for everything that happens on the moving space station or in the flying rocket. All processes taking place on these moving objects will appear slowed down for our observer.

Characteristically, there are situations where time dilation is mutual. For instance, if there are two observers drifting through space, each on his or her own space station, and if those two space stations are in relative motion, then for each observer, the time in the other space station appears to run slower than for himself. (If that already sounds like a paradox to you, you might want to read the spotlight topic The dialectic of relativity.)

With the help of time dilation – often abbreviated to “moving clocks go slower” – one can try to explain what happens to the twins. No wonder the travelling twin ages less! After all, the twin on Earth can invoke time dilation: Moving clocks go slower, and so do the clocks of the moving twin. On these slower-moving clocks – and, by extension, in the whole spaceship – less time passes than on Earth, in other words: when the travelling twin returns, he is younger.

No paradox so far. But why can’t the travelling twin turn the tables on her sibling? After all, motion is relative. Why can’t the twin in the spaceship define herself as being at rest? From that point of view, it would be the Earth that moves away before returning to the spaceship. And if that is so, couldn’t the travelling twin apply time dilation (“moving clocks are slower”) to everyone who remained on Earth? By that argument, shouldn’t it be the humans on Earth that are younger than expected once the twins are reunited? If both twins are on an equal footing, then each one should be allowed to consider herself at rest and invoke time dilation. But in the end, when the twins meet again, only one of them can be right – then, there cannot be any ambiguity: either the one twin is younger, or the other (or, of course, both twins’ arguments are wrong, and they have aged exactly the same). A contradiction – a twin paradox?

The importance of inertial observers

To resolve the contradiction, a closer look at time dilation is needed – in which situations do moving clocks indeed go slower? In the above text, the key criterion was hidden in parentheses: For the dictum “Moving clocks go slower” to hold, you must be an inertial observer. The example of freely floating space stations above gives a flavour of what this qualification means: In an inertial reference frame, all objects are perfectly weightless. For such observers, an object upon which no external forces act (for instance, that is neither pushed nor pulled) either remains at rest or moves with a constant speed along a straight line.

There’s the litmus test for each twin: Is she an inertial observer, and thus entitled to apply the time dilation formula, concluding that moving clocks go slower?

An unfortunate complication: The twin that remains on Earth is no inertial observer. She’s in a gravitational field in which objects fall down instead of remaining at rest. There are two possible ways to proceed. Either one can use Einstein’s theory of gravity, general relativity, and calculate how the gravitational field influences the twin on Earth. The result is that, in the given situation, the Earth’s gravity does not make an appreciable difference. If we ignore Earth’s gravity and treat the twin on Earth as an inertial observer, our results regarding the relative aging of the two twins will be correct, give or take a few fractions of a second. If we choose situations in which the twins eventual age difference is counted in years, gravity will not matter.

Alternatively, one could re-define the situation by having the non-travelling twin wait not on Earth, but in a freely floating space station in deep space, far away from any massive objects. That would definitely make her an inertial observer.

In both cases, the result is that the non-travelling twin has the right to apply the simple time dilation formula, and to conclude that her travelling sibling will be younger when they meet again.

What about the travelling twin? She’s not an inertial observer, either, at least not the whole time. If she simply would simply coast along with constant speed along a straight line, she could never return to Earth (or, in the alternative version, to the other twin’s space-station). In order to return, it is crucial that the travelling twin either come to a stop and accelerate towards Earth or, alternatively, fire her engine to force her spaceship onto a tight curve to point it back towards Earth. In both cases, the travelling twin feels the acceleration – decelerating, her body feels a pull in the direction offlight, re-accelerating, she is pressed into her seat, in flying a turn, she is pulled sideways. Acceleration is inevitable – and while she is accelerating, the travelling twin definitely isn’t an inertial observer. For instance, during a braking phase, objects afloat inside the spaceship’s cabin will not just float or move with constant speed – they will be accelerated toward the front of the spaceship. And in contrast with the twin on Earth, there is no slight redefinition that will do away with these acceleration phases. There’s no way around it: At least for some of the time, the travelling twin isn’t an inertial observer.

Thus the apparent paradox is resolved. The twins are not on an equal footing. The accelerated twin cannot just apply the simple time dilation formula, while her sibling on Earth can. The latter twin’s conclusion that the clocks of the travelling twin run slower, and that the travelling twin is thus younger when they meet again, is valid. (So what role does the acceleration play in this? Find out more in the spotlight topic Twins on the road.)

Further Information

The basics of special relativity – the proper theory to answer all questions about these twins – can be found in Elementary Einstein in the section Special Relativity.

Related Spotlight topics on Einstein-Online an be found in the section Special Relativity.

This is just a draft, I will sort it when my burnt fingers recover!