Page last updated 4/12/2023.
I am not a physicist having undergone only VCE Physics, I know approximately nothing, so anything I say on the physics may not be accurate. If you identify anything wrong, however minor, please email me at nhtnhanbn@gmail.com.
If it seems like a way to procrastinate on preparing for a final exam it is.
The Study Designs, examinations and teaching of the subjects offered as part of the Victorian Certificate of Education, administered by the Victorian Curriculum and Assessment Authority and often completed by year 12 as both a graduating and matriculating process, are often not perfect. Physics is no exception.
One of the concepts studied and tested as part of Units 3/4 VCE Physics, undertaken generally in year 12, is Einstein’s theory of special relativity. However, conspicuously (to me) absent from both the 2017-2023 and the 2024-2027 Study Designs is explicit prescription of relativity of simultaneity. Minkowski diagrams are also missing and they also should not be (they would be a good aid to many of the explanations to follow, but are outside the scope of this piece) - nevertheless, I consider neglecting relativity of simultaneity to be the greater crime.
Essentially, relativity of simultaneity states that two events in two different locations observed to be simultaneous in one frame of reference will be observed to be not simultaneous in another frame of reference that is in relative motion. This is a necessary consequence if the vacuum speed of light has a constant value for observers in all frames of reference.
Albert Einstein (or a translator) justifies the relativity of simultaneity quite well through the thought experiment in Chapter IX of his book Relativity: The Special and the General Theory. It can be read as a standalone chapter without needing to struggle through the difficult preceding chapters.
Much of what is said in this section is original (in other words, I made it up and cannot be trusted) and potentially incorrect, the more important section is the next.
To assume an absoluteness of simultaneity, that is an absence of relativity of simultaneity, would be a fatal misconception.
This will be in many ways symmetrical to Einstein’s thought experiments. Suppose Nhan and Scheelite are standing on opposite ends of a railway embankment, and a train is travelling parallel to the embankment, away from Nhan and towards Scheelite. As the midpoint of the train passes the midpoint of the embankment (halfway between Nhan and Scheelite), lightning strikes the midpoint of the train which at that time coincides with the midpoint of the embankment.
Consider the frame of reference of the train. Nhan is moving away from and Scheelite is moving towards the train. After any period of time after the lightning has struck, Nhan is a greater distance from the midpoint of the train than Scheelite. It is clear that Nhan sees the lightning stroke later than Scheelite does. The two events, the light emitted from the lightning stroke reaching Nhan and the light emitted from the lightning stroke reaching Scheelite, are not simultaneous.
Consider the frame of reference of the embankment. Nhan and Scheelite are both stationary. After any period of time after the lightning has struck, Nhan is still an equal distance from the midpoint of the embankment as Scheelite. It is clear that Nhan sees the lightning stroke at the same time as Scheelite does. The two events, the light emitted from the lightning stroke reaching Nhan and the light emitted from the lightning stroke reaching Scheelite, are simultaneous.
This presents an apparent paradox. It is clear that accepting that simultaneity is relative is necessary to resolve this paradox. Thus, we can determine a single, correct and singularly accepted, answer to Question 2 of Detailed study 1 from the 2016 VCE Physics exam, and Question 3 of Detailed study 1 from the 2010 VCE Physics exam 1 (June) set by VCAA.
This is a classic example of an apparent paradox under special relativity.
A ladder has length that just fits inside the length of a garage. The ladder travels, left to right, parallel to the ground through the garage which has open doors on either end. As the ladder passes through the garage, the doors close, simultaneously in the frame of reference of the garage, and trap the ladder.
We say the ladder fits inside the garage if at an instant, both ends of the ladder are simultaneously within the garage. Does the ladder fit inside the garage?
Consider the frame of reference of the garage. The ladder, being in motion, is observed to have its length contracted. The doors close simultaneously. In the frame of reference of the garage, the ladder will fit inside the garage.
Consider the frame of reference of the ladder. The garage, being in motion, is observed to have its length contracted. In the frame of reference of the ladder, the ladder will not fit inside the garage. In fact, the two doors closing, which are simultaneous in the garage’s frame of reference, are not simultaneous in the ladder’s frame of reference. The right door closes first and the right end of the ladder hits it. By the time the left door closes, the garage is stationary in the ladder’s frame of reference and the ladder indeed fits (just). It should be realised that in fact, the ladder before and after collision with the right door are two different frames of reference.
The observations made in all frames of reference are fully valid. Length contraction is symmetric between inertial frames of reference. There is no real paradox.
Say Nhan is stationary on Earth and Scheelite travels at constant velocity along a straight path connecting the Earth and the star Wolf 1061. They synchronise their clocks as Scheelite passes the Earth. Assume that Wolf 1061 is stationary relative to Earth at a distance of 14 light years away. Whose clock is slower?
Consider the frame of reference of Nhan. As Scheelite is in motion relative to Nhan, Nhan observes Scheelite’s clock to undergo time dilation; it is slower than Nhan’s clock. Hence, when Nhan observes Scheelite reach Wolf 1061, Nhan will observe that Scheelite’s clock is behind Nhan’s clock; Scheelite has experienced less time.
Consider the frame of reference of Scheelite. As Nhan is in motion relative to Scheelite, Scheelite observes Nhan’s clock to undergo time dilation; it is slower than Scheelite’s clock. Hence, when Scheelite reaches Wolf 1061, Scheelite will observe that Nhan’s clock is behind Scheelite’s clock; Nhan has experienced less time. In fact, where in Nhan’s frame of reference, the two events Nhan observing Scheelite reach Wolf 1061 and Scheelite reaching Wolf 1061 are simultaneous, in Scheelite’s frame of reference the two events are not simultaneous.
The observations made in all frames of reference are fully valid. Time dilation is symmetric between inertial frames of reference. There is no real paradox.
Also note that when Nhan observes Scheelite to be 14 light years away (at Wolf 1061) is not simultaneous with when Scheelite (not at Wolf 1061) observes Nhan to be 14 light years away.
As an aside, the classic full twin paradox can be resolved using special relativity alone, without relying on “acceleration” (in the sense of being in a non-inertial frame of reference). It is worth noting that it is a misconception that dealing with acceleration requires general relativity - in fact, general relativity is only required when dealing with curved spacetime and special relativity can account for acceleration just fine, it is just more complicated to do so. Here is a scenario to consider that involves no acceleration. Say Scheelite is always at constant velocity. Scheelite is already at velocity when they pass the Earth, at which point Scheelite and Nhan set their clocks/ages to the same time (as in the scenario just considered), that is Scheelite does not accelerate as they leave Earth. As Scheelite passes Wolf 1061, Wolfram, travelling at the same speed in the opposite direction, passes Wolf 1061. As the positions of Scheelite and Wolfram coincide, Wolfram sets their clock/age to equal Scheelite, that is Scheelite/Wolfram do not accelerate around Wolf 1061. As Wolfram passes Earth and the positions of Wolfram and Nhan coincide, they make their observations, that is Wolfram does not accelerate as they arrive at Earth. There is no acceleration in this scenario, but Scheelite and Wolfram are different frames of reference. One reality must be true, the solution is left as an exercise for the reader.
Suppose that two clocks in two different locations in the same inertial frame of reference, being stationary relative to each other, have been synchronised. They are synchronised in their shared frame of reference. Then because of relativity of simultaneity, the clocks are not synchronised if observed by an observer at motion relative to them.
Suppose that two clocks are situated on opposite ends of a train and are synchronised in the train’s frame of reference. The train passes at constant velocity, travelling left to right, a post stationary on a railway embankment. As each clock passes the post, they record their time and the post’s clock records each time.
Consider the frame of reference of the train. Its clocks are synchronised. It records a certain time difference between its clocks passing the post. The train also observes the post’s clock running slower because the post is in motion relative to the train, meaning that it observes the post record a time difference less than the time difference recorded by the train.
Consider the frame of reference of the post. It records a certain time difference between the train’s clocks passing it. The train also observes both of the train’s clocks running slower because the train is in motion relative to the post. However, we know that in the post’s frame of reference the train’s clocks are not synchronised due to relativity of simultaneity, with the right clock behind the left clock. Therefore, although the post observes the time running slower for the train’s clocks, it also observes the train record a “time difference”, defined by how we have constructed this scenario, greater than the time difference recorded by the post. The time difference recorded by the post is considered the “proper time”, being recorded in the same position in its frame of reference.
The above scenario can be used to construct a thought experiment that, in my personal experience, allows a quite intuitive justification for length contraction (once we accept relativity of simultaneity).
We realise how important relativity of simultaneity is in special relativity. I suspect that it is not sufficiently understood by too many close to VCE Physics. Much of what I had to learn about it, to resolve my understanding of the other parts of special relativity like time dilation and length contraction, I had to do my own reading for owing to the lack of specifically VCE resources that deal with it. The consequences of ignoring such an important idea that underpins so much of special relativity in VCE Physics are unacceptable.
Whilst often we can ignore relativity of simultaneity by getting by with accepting definitions or formulas of time dilation, length contraction, postulates and whatever else (although it is easier to trip up without consideration of relativity of simultaneity), there appears to be a class of questions for which ignoring relativity of simultaneity can be actively fatal, and all of students, VCAA examiners and trial exam setters have been killed by it.
Take the example of Question 3 of Detailed study 1 from the 2008 VCE Physics exam 1 (June) set by VCAA. The exam report indicates that all answers (except No Answer which is incredibly unfair if incorrect answers were accepted) are considered “correct”, with no comment. Clearly that cannot be true, as contradictory observations should not be made in the same frame of reference. Can we identify the correct answer? The exam report was amended in 2011, which may or may not relate to this question.
If Fred receives both signals of light simultaneously, the motion of the train meaning that the light has travelled further from Alan’s original position than from Bob’s to reach Fred, I would suggest the answer should be that Nancy sees the light from Alan first. Note that 45%, a plurality, of students obtained the answer agreeing with me.
Take another example of Question 18 of Section A - Multiple-choice questions from the 2022 trial exam published by the Science Teachers Assocation of Victoria. It is identical to 2016 VCAA exam Detailed study 1 Question 2, where Barry becomes the Earth, Anna becomes the captain on the spaceship, and the space lab becomes the space station. The answer given by STAV, that the captain observes the Earth and space station receive the radio pulse at the same time, disagrees with VCAA, suggesting that STAV has not appreciated the relativity of simultaneity which makes their proposed answer necessarily incorrect.
Another VCAA question requiring relativity of simultaneity was Question 8 of Detailed study 1 from the 2004 VCE Physics Pilot exam 1 (June). In 2004 two different cohorts of VCE Physics students took different exams on different study designs. This exam was the new study which introduced optional study of Einstein’s relativity. The scenario is obviously similar to the previously referenced Einstein thought experiment. The exam report clearly indicates that the examiners understood the necessity of applying relativity of simultaneity. But their answer appears to be opposite to Einstein’s result. From my reading of the report I am not sure why.
Take Question 2 of Detailed study 1 from the 2016 VCE Physics exam which we have previously mentioned. The answer given by VCAA is correct. The question was the second-worst answered question of the detailed study, and the correct answer was only the second-most popular selection for that question among students.
The point is that accepting relativity of simultaneity should be something taught to every VCE Physics student. The Study Design does not necessarily need to explicitly mention it, although it would be good. However, it is a natural consequence of what is included in the Study Design. As a concept, I think it is not more difficult than length contraction, time dilation or relativistic energy, yet it helps to understand or at least accept the rest. Is it necessary, for the purposes of VCE Physics, to fully understand all its applications and implications, some of which are touched on above? Absolutely not. But knowing that it is real means that we can hand-wave away any apparent paradoxes, misunderstandings, asymmetries. It gives us permission to accept that the observations from different frames of references can be different, and so obey the fundamental rule of answering relativity exam questions. Consider only the frames of reference relevant to the question.