### Demystifying special relativity Imagine two surveyors assigned to survey layout of a building: Alice and Bob. To survey, you have to have a compass. Alice's compass is aligned to magnetic north while Bob has a fancy compass whose compass points to the true north of the Earth. It's obvious to see that when they survey, they will come up with different _data_ for where things are. For Alice something that's pure North might be North-West for Bob. Question is: who is right? Well, both are. The data they got depended on their reference system. A much simpler example is when people use different units. What is 1 mile for you is 1.6 kms for me. Same thing, different data. One key point to note is that even though Alice and Bob might disagree on specific locations of objects they're surveying, they will always agree with the distance between two objects. So, distance is an invariant in this case, which means different surveyors will agree on the distance measured irrespective of their reference systems. #### Postulates of Relativity Special relativity is an elaboration of the idea above with just one experimental fact: the speed of light is the same in all reference frames. Actually, all special relativity can be derived from just two postulates: >**Postulate 1**: laws of physics remain the same in all inertial reference frames >**Postulate 2**: the speed of light remains the same in all inertial reference frames Here, the term "inertial reference frame" (IRF) means a non-accelerating point of view where you can be seen as being at rest. So, for example, sitting on a moving train is IRF because an observer moving at the same velocity as the train will see you at rest. Postulate 1 simply says that the results of your experiments should not depend on which IRF you're in. This means all IRF are equivalent as far as the laws of physics are concerned. The postulate matches our intuition because when two trains are crossing, sometimes it appears that we're at standstill and the other train is moving really fast. Similarly, in a closed spaceship you would not be able to tell if you're moving or at rest because movement or being at rest is always in reference to something else. Absent a reference point, the concept of velocity is not even meaningful. **So, you can't say you're moving or at rest. You always say you're moving or at rest as _compared_ to something else.** Postulate 1 is important for special relativity because it essentially makes all IRF "equally valid". If we could tell which IRF we're in, we could designate one particular IRF as special and measure results of experiments in that IRF. But because no IRF is special, universe _forces_ us to accept consequences of all IRFs being equally valid. Postulate 2 is not intuitive at all. In fact, it's counter-intuitive. Our typical intuition is that velocities add up. If someone is walking in a train at velocity $u$ (as compared to train) and the train is speeding up at velocity $v$ (as compared to us), when we look at the person, we see her motion as having velocity $u+v$. But, this is not in true for the speed of light. If someone shoots a beam of light on the train that's moving at the velocity $v$ (as compared to us), we will not see the speed of light as being $u+c$, but rather $c$ only.  (via [here](http://abyss.uoregon.edu/~js/ast123/lectures/lec08.html)) It's hard to wrap your head around why is this so. It seems like a fact of the universe that we simply have to deal with. It's obvious that the speed of light cannot be zero or infinite because then everything will happen all at once or nothing will happen at all (which is the same thing). So the speed of light had to be _something_, but why did it have to be same in all reference frames? Nobody knows. Perhaps there's some deeper reason why the speed of light should be a constant in all reference frames. We know this is the case because all experiments we've done so far have confirmed this. But this is not derivable from any theory or first principles. So far, I haven't come across any. It seems like an #openquestion. #### What is time in Relativity? We're used to the idea of an absolute time. If it's 5pm for me, it's 5pm for you as well. Of course, we may be in different time zones but we feel like there's a global clock ticking in the universe that advances all clocks simultaneously at the pace of 1 second per second. This intuition of time passing at a constant pace everywhere simultaneously directly feeds into our sense that one hour passing for me means one hour passed for my friend as well. In other words, we all age together and reach the same future time together. However, the notion of universal clock is wrong. **Time is what we read off the clocks, and all clocks measure repetitions of some periodic motion (converted into an appropriate time unit).** For example, here's how the popular Quartz based clocks work. Quartz crystal when struck vibrates at a particular resonant frequency. Quartz, being a piezoelectric material, generates pulses of electricity when it vibrates. So quartz based watches count the number of such pulses (and hence number of vibrations). For each 32768 counts/vibrations, the clock counts one second. So, **time we measure fundamentally depends on motion. In fact, all periodic motion can be used to measure time duration**. This gives clear insight into why time is relative. Imagine you've built a clock that measures time by keeping track of how many times a photon has bounced off the two mirrors across each other (that are L meter apart). If you're stationary with respect to the clock, you simply measure duration between ticks as $2L/c$. However, if you're moving with a velocity of $v$, you'll measure duration between ticks as $2D/c$.  It's clear that since light has to travel a longer path in a moving clock (_from your point of view_), the duration of ticks in a moving clock will be longer as compared the duration of ticks in a clock stationary as compared to you. This is why moving clock runs slower. That is, for every 10 seconds in a clock stationary next to you, <10 seconds would pass in a moving clock (since each second takes longer to tick). **This consequence is called time dilation** and it obviously isn't just limited to photon clocks. Since all clocks depend on motion, all clocks will slow down (including the biological clocks). So a moving person indeed would age less when viewed from the point of view of a stationary person. ### Simultaneity is relative We take simultaneity for granted here on Earth. If there's a birthday party at the same time as screening of our favorite show, we have no trouble saying that those two events are simultaneous. Also, and importantly, we believe our friends will also see that these two events coincide in time and hence may be accommodating to shift their party to some other time. However, in reality, simultaneous events for me are not simultaneous for you if we're moving with a relative velocity. For example, imagine I'm sitting in the middle of a ledge with two torches at the ends. If these torches are on, I'll receive their light/photons simultaneously (since I'm sitting in the middle, so distances from both torches is equal). However, if the ledge+me are moving relative to you, then since one end is closer to you than the other end, you will see their light/photos received at different moments. **So, what events are simultaneous for me are not simultaneous for you. And we both are right. ** The word "simultaneous" loses its meaning in relativity because time and duration between events depends on which reference frame you adopt. For example, assume one day we get to see Jupiter and Mercury exploding on the same day on Earth. For someone on Mars, Jupiter would explode first and then Mercury while for someone on Venus, Mercury would explode first and then Jupiter. (This is because light takes different time to reach different places). So who is right? Did the planets explode at the same time or at different times? Well, all are right and answer to simultaneity question depends on whose reference frame you adopt. But, in the example above, note that even if someone at Venus comes to know that Mercury has exploded before someone knows that at Earth, that Venetian can never actually tell that to Earthling before the actual event is observed by the Earthling (because no message can travel faster than light). So, for the Earth reference frame, there's no way to know that Mercury has exploded before the light reaches from Mercury. And in that sense, it's not fair to say that the event happened _before_ at Venus. Hence, simultaneity of events and the "present" moment is only validly applicable when the distances are short (nearby areas at say the earth-like diameters, where our perception of time duration is much less than what we can notice consciousness) and the velocities are much less than the speed of light). **There is NO universal simultaneous moment because there's no absolute universal clock ticking**. As the philosopher of science David Mermin [says](https://en.m.wikipedia.org/wiki/Rietdijk%E2%80%93Putnam_argument#Andromeda_paradox): >That no inherent meaning can be assigned to the simultaneity of distant events is the single most important lesson to be learned from relativity. #### World lines with constant speed of light All normal, weird, "paradoxical" consequences of special relativity can be derived from the fact that the speed of light is constant in all reference frames. To see how, first let's draw a reference frame. We will represent time on the Y axis and space on the X axis. For convenience, we're only using one dimension of space but in reality we have three dimensions of space and one dimension of time. Since we can't represent 4D on a screen, let's make do with a 2D graph. It'll help clarify all essential concepts of special relativity.  (via [here](http://www.madsci.org/posts/archives/2002-01/1011381196.Ph.r.html)) Natural units of distance/time. In such graphs, we keep the speed of light as 1, which means distances and time are both measured in units of light speed. For example, if one tick vertically is 1 second, then one tick horizontally will be one light-second (the distance light travels in 1 second). Since, the speed of light is 1, the world line of light makes a 45 degree angle and all other objects (with mass) travel less than the speed of light and hence make an angle less than 45 degree angle. In such graphs, a person can be said to be at rest while the other is moving with a velocity $v$ (or vice versa). So we have easy transformations between two moving reference frames. These transformations are called Lorentz transformations and they satisfy the property of leaving all 45 degree lines unchanged (while transforming all other lines.) These 45 degree lines have to be unchanged because the speed of light is 1 in all reference frames, so no matter which reference frame we change to, we should always observe the light going at the speed of 1.  (via [here](https://commons.wikimedia.org/wiki/File:Animated_Lorentz_Transformation.gif)) In the animation above what's happening is that the points that are on vertical axis (stationary frame) are shifted to the left as we make some other point the vertical axis (stationary frame). Notice how this transformation changes the time duration between events in the original frame and how originally simultaneous events (having the same vertical value) are no longer simultaneous in the transformed co-ordinates. However, notice how the two 45 degree lines remain unchanged in these transformations. This is simply because no matter which reference frame you adopt, light always travels at the same speed. ### Past, Future and Causality Because light takes finite time and nothing can go faster than it, around each moment/event (at a particular location in space and time), we can imagine a set of events that are capable of influencing it (say, by sending a beam of light) and the set of moments it is capable of influencing (by sending a beam of light). Outside these set of events, all other events are causally disconnected from the current moment. The sets of events that can be impacted are called future light cone and the set of events that can impact current event is called past light cone.  (via [here](https://www.youtube.com/watch?v=wDwXOH16USg)) It's interesting to note that the causal ordering of events can change depending on the perspective. For example, in the following worldlines, the ordering of events is 1, 2 and 3 for the observer at rest but 2, 1 and 3 for the moving observer. ![[causal-ordering 1.png]] (via [here](https://www.youtube.com/watch?v=wDwXOH16USg)) However, note that both observers agree that event 1 happened before event 3. This is because events 1 and 3 are within past/future cones of each other, which means that all observers will agree on the ordering for them (since 1 can impact 3 by sending a signal slower than speed of light). On the other hand, event 2 is outside the light cone of 1 and 3 and its ordering will depend on how the observer is moving relative to it. ![[Screenshot 2021-09-24 at 2.56.01 PM.png]] (via [here](https://www.youtube.com/watch?v=wDwXOH16USg)) #### Invariants in special relativity As we have seen so far, time duration and distance between two events are relative concepts in relativity and their measurement depend on the frame of reference. However, just like initial example, all surveyors agreed on distance between two objects (irrespective of their reference systems), there is an analogous concept in relativity which all observers would agree upon. Such measures that don't don't change between reference systems are called invariants. In special relativity, one such invariant is the speed of light $c$ that remains the same in all reference frame. Another invariant that remains the same is **spacetime interval**. For two events, it is defined as: $(\Delta s)^2=(\Delta t)^2-(\Delta x)^2$ Where $t$ is time dimension and $x$ is space dimension. If there are three spatial dimensions, replace $(\Delta x)^2$ with $(\Delta x)^2+(\Delta y)^2+(\Delta z)^2$. What it means for spacetime interval to be invariant is that _both_ stationary and moving reference frame will measure this interval to be same when they plug in their measurements of time and space for the two events. Because space time interval is an invariant, this can be used by a stationary observer to imagine time and distances from the point of view of the moving observer. For example, in the following diagram an observer at rest observes another observer moving from point A to B. The question is: how much time would have passed from the moving observer's point of view and how much apart are two events? ![[spacetime-interval.png]] The simple way to calculate as follows: - Measure $(\Delta t)$ and $(\Delta x)$ as you see in the graph (for points A and B). - These lengths are from the stationary observer's point of view - Plug these numbers into the formula of spacetime interval to calculate $(\Delta s)^2$ - Since a moving observer feels herself at rest in her own reference frame, we can imagine her not traveling any distance in that reference frame, keeping $(\Delta x)=0$. - Plug in $(\Delta x)=0$ into the spacetime interval equation and use the value of space time interval that we got above - This gives us $(\Delta t)$ from the perspective of moving observer being at rest - **This is called proper time**, which is the time felt/observed by the moving observer from _her own_ perspective. - ** We can immediately see time dilation from this**. Notice how the value of spacetime interval reduces if $(\Delta x)>0$ since this value is getting subtracted. Since spacetime interval is proper time of a moving observer, the duration as measured by moving observer in her reference frame will always be less as compared to duration as measured by stationary observer. - We can do similar calculation to calculate **proper length** which is the length a moving observer will measure from her _stationary_ perspective. We simply keep $(\Delta t)=0$ to measure distances between events at simultaneous points of time (and we reverse the sign in calculating spacetime interval also). - This also shows how length contracts for a moving object since whatever is simultaneous in my stationary reference frame (interval with $(\Delta t)=0$) if viewed from a moving reference frame is not simultaneous in your reference frame. So when you calculate spacetime interval, you subtract $(\Delta t)^2$ from $(\Delta x)^2$. When I calculate my proper length using the same spacetime interval and plug in $(\Delta t)=0$, my $(\Delta x)$ will have to be bigger than what you measured for us to arrive at the same invariant value of spacetime interval. Hence, you'll always measure lengths of things moving in my reference frame smaller than I do. **Hence length contracts.** #### Surprising and counter-intuitive phenomena predicted by special relativity ##### You age less if you move somewhere and come back to my rest frame The so called "twin paradox" says that if you move away from me and either travel a large distance or at a fast speed or both and then come back and meet me, I would have aged more than you. The hard thing to digest in this "paradox" is the concept of missing years. Where did my "extra" years disappear for the traveling person? The fallacy here is that you implicitly assume that there is a universal clock that ticks the same for everyone. As we've seen, that's not true. The math for this aging "paradox" works out but I had a hard time intuitively understanding how can it be possible. What drove home the understanding for me was to imagine what would the stationary and traveling person "see" if each was sending the other a photo or a movie at regular intervals from their respective locations at a specific interval (say each month or year). The following video makes it very clear what would be seen by the two people: <iframe width="560" height="315" src="https://www.youtube.com/embed/h8GqaAp3cGs" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe> Or, see the following world line from [this excellent article](http://www.physicsmatt.com/blog/2017/1/18/the-twin-paradox-in-special-and-general-relativity): ![[twin-paradox.png]] Essentially, the stationary person will (overall) see the moving person behaving in slow motion while the moving person will (overall) see the stationary person behaving in fast forward. Though, as the video above shows, in different parts of the journey, they will see both speed up or slowed down. However, the overall perception will hold as I've written above and therefore the moving person would have aged less than the stationary one. ##### An object can _both_ fit and not fit in a particular gap depending on the perspective If an object is moving fast, it contracts in length with respect to stationary observer. So if there is a gap (like a garage) in stationary frame, depending on how fast the object is moving, it can always fit in the gap (because at the speed of light, length contracts to zero). However, from the moving object's point of view, its the gap that's moving and it is stationary and hence the gap should contract and it shouldn't be able to fit inside the gap. So, can the object fit or not fit in the gap? Because the idea of simultaneous events depend on the reference frame, the answer to these questions also depend on the reference frame. The following video does a good job explaining this "paradox": <iframe width="560" height="315" src="https://www.youtube.com/embed/YVhI45_WzJ4" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe> The essential idea is that fitting inside the gap requires both ends of the object to be contained inside the gap simultaneously, which happens in one frame but doesn't happen in another frame. So, no paradox really. Two observers are defining simultaneity different and hence they're looking at _different_ events in their world line when they're talking about whether something fits inside the gap or not. ##### Traveling to the edge of observable universe within a human life time We generally assume that it's impossible to travel more than X light-years distance in less than X years because nothing can go faster than light. Which means for a human with expected life span of 80 years, it can be assumed that they can never travel farther than 80 light years which is just a few stars within our galaxy. This assumption certainly confines humans within our own galaxy since the nearest galaxy Andromeda is 2.5 million light years away. However, this assumption is not valid. When an object is traveling, it can be assumed to be at rest and other object can be said to be traveling. Since moving lengths contract, in reality, for a really fast object, the distance to Andromeda can be covered within a short experienced time. We can also see this effect from time dilation point of view. What will be 2.5 million years from Earth point of view, will be just a few years from fast moving spaceship point of view. For example, at 1g constant acceleration (which our body can handle), [we can reach Andromeda within 57 years](https://news.ycombinator.com/item?id=28503623). This effect of time dilation/length contraction is actually what enables us to detect fast decaying muons on Earth surface. Their lives are extremely short (picoseconds order) and even at near speed of light, they can only travel a few meters before decaying and hence never be able to reach the surface of earth from the atmosphere. However, we regularly detect them and that's because at their speeds, time dilates for them they live a lot longer in our frame. This video explains it well: <iframe width="560" height="315" src="https://www.youtube.com/embed/rVzDP8SMhPo" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe> Hence, space travel using constant acceleration to take us to high velocities is the best way for humans to reach far reaches of the galaxies. However, even if the traveling person can reach Andromeda within her lifetime, back home at Earth 2.5 million years would have passed. >At a constant acceleration of 1 _g_, a rocket could travel the diameter of our galaxy in about 12 years ship time, and about 113,000 years planetary time. ([source](https://en.wikipedia.org/wiki/Space_travel_using_constant_acceleration)) Constant 1g acceleration gets near speed of light within 1 year and hence both length contracts and time dilates. This enables us to travel not just to Andromeda but actually [the edge of observable universe within our lifetime](https://www.universetoday.com/129086/far-can-travel/). How cool is that!