What does it mean to say, Light-time?

This is gonna be our last topic on the terrific astronomical measurements and it’s the most enjoyable & tasteful topic of all (Why? Because, it has Light). What you understand here, paves the way for you to grasp a fraction of Einstein’s mind (a fraction that is). And, you should pardon me (alot of times) along the way, that mostly I’ll be saying, “we’ll see about that in a later post”. Because, today we’ll be going along blindly without investigating much in, “Why’s that so?”. We’ll just digest that, that’s the way it works, and investigate one by one, in the near future. Don’t you worry, all those things will come handy in due time, as this terrific stuff needs a few posts…

Okay, it’s actually called out as light-year. But, you can have light-seconds, light-minutes (of course we have), light leap-years (Meh..??) if you want. It’s just an unit. Ain’t it? When you’ve finished playing around with the units, be sure that you’ve plugged in the right numbers responsible for such a transformation to happen. Roughly (in your language), when you change an inch to a centimeter, make sure that you’ve included the “2.54” factor.

Alright, so what’s the big deal about light? If you’re a beginner to Physics, then you might believe in a lot of statements like this one, “The speed of light is something very special in our universe, that no one can reach the speed”. You’re quite right in a sense that it’s the cosmic speed limit. But as you become an amateur, you’d easily figure out that what you previously assumed was not really true. Before we get into light and give an introductory to spacetime and stuff, let’s have a look at light-year, which is what this post was intended to address…

Light-time is distance…

Duh, recall high school physics. Roughly, the velocity with which the object travels, times the time interval gives the distance it has covered within that interval. So, light-time is nothing but the distance covered by light, within that specific time unit you’re using. For instance, it takes 1.26 seconds for light to reach Moon from Earth. Hence, the moon is at a distance of 1.26 light-seconds. Below, is the Earth and the moon (to scale) where a light pulse is sent from Earth (Now, that’s realistic)…

It’s worth noting that whenever we use “light-time” to measure distance, we always make use of the speed of light in vacuum c, which is 299,792,458 m/s (with some good accuracy). There are other crazy examples, such as 1 meter equals 3.3 light-nanoseconds, the length of Earth’s equator equals 134 light-milliseconds, etc. Before getting into the reason to your question, “Why’s that so?, Why’s light chosen?” etc., let’s watch Henry’s video on “How far is one second?”…

I’ll brief that out now. Just like you say, “It takes 5 minutes for me to reach the shop from home by walk”, the physicists say, “It takes 8.3 minutes for light to reach Earth from Sun”. The use of light-time to measure some bloody distance may look awful, as if light has been given some special preference. Nope, this conclusion is wholly based on Einstein’s great amalgamation in the 20th century, that space and time, both are different aspects of exactly the same thing, a wonderful universal mathematical framework called the spacetime. It’s the stage where every single event takes place in the universe. It’s also the stage where different observers agree with the distance. “What’s so special about this framework? What’s an observer and how many are there? Does this c has anything to do with it?, etc.” will become much more clear in the future.

All you need to know now is this crude analogy which goes like this. If you’re to cover more space, then it takes less time and the vice versa. Say you’re traveling in the Northerly direction in space (in some vehicle of yours). If your path is slightly tilted towards north-east, you’d be covering some portion in the east, and some in the north, as shown above. That’s what Pythagoras says. In the figure, d represents your distance along the north-east direction. The dcos\theta and dsin\theta are the portions of distance contributed by your motion to the north and east directions. The cosine & sine are just the projections of your distance with the respective directions. Let’s check whether they give your distance or not. (You do remember Pythagoras. Right?)

d^2=(dcos\theta)^2+(dsin\theta)^2

=d^2(cos^2\theta+sin^2\theta)

which is hopefully d^2 again. As a conclusion, Pythagoras says that the average distance covered by you along the north-east direction will be the sum of squares of the contributions of your motion to each direction (If you’re also including up/down, that contribution should also be plugged into the expression). So, we’ve done our “thing” in Euclidean (flat) space.

“Crude” introduction to spacetime…

Now, why did I declare the “north-east” analogy (above) to be crude? Because, it fails when the speed of the object attains a certain fraction of the speed of light. And that’s because, spacetime doesn’t make use of Pythagorean theorem. Instead, it makes use of a modified version of Pythagoras, which our dear mathematicians call, the hyperbolic space. As there’s no absolute position in space or absolute interval in time, we need to construct something that’s agreed upon by everyone. That’s what this four-dimensional framework is gonna do. For now, let me give an abstract explanation about the non-absoluteness of a certain position in space. I can convince you in two ways.

Who are these observers?

Firstly, lemme ask you a question, “How can you declare something to be at an absolute rest (i.e) not moving?” You say, “Well, that’s easy. I’ll sit on my chair and I’m at rest”. Now, I can say that you’re moving at 29 km/s around the sun (as you’re in Earth). You say, “Alrighty then. I’m going to outer space and sit somewhere peacefully (??) between Sun & Earth”. Now, I can argue that you’re moving with the solar system around the galactic center at 220 km/s. “Okay, I’m going elsewhere, outta this freaking galaxy itself”. In such a case, I can say that the space itself is expanding and new voids are coming and filling up. You breathe out a “sigh”…

We’ve concluded that you can speak of rest only relative to someone else. First, you were signifying that you were at rest relative to Earth, then relative to the Sun and so on. It also follows, (as there’s no absolute resting) that when you’re at rest, I can prove that you’re in motion. All I have to do, is move relative to you. And, you’ll be moving relative to me. That’s relativity. There’s no absolute motion. Moreover, these scientific people need “numbers” to calculate things. So, whenever you and I are about to be calculated, we’re assumed to carry a set of co-ordinates with us (i.e) the x, y & z axes. Formally, these are called the frames of reference and we’re called the observers. And, you do all your calculations with your numbers in these frames. Observers are hypothesized to carry sophisticated set of scientific instruments for measurements in their frames.

And, this is exactly what Galileo thought. He and Newton, both were convinced that space isn’t absolute. But, they still stuck to the “false-fact” that time was absolute, which was discarded only when Einstein came into the play. The abstract statement is this, “You haven’t achieved relativistic speeds in your whole mortal life. If you did so, you’d come to the same conclusion”. If you wanna know now, it requires some (not so tough) math (as usual, it’s for the future)…

What have I drawn here? This is called a spacetime diagram where physicists have the weird convention of plotting space along the horizontal and time along the vertical. Always keep in mind, this isn’t a graph. Nothing is dependent of anything. Both space & time are interdependent things. “Both are different” is just an illusion because, you’re a non-relativistic observer. This whole universe has been constructed in such a way that both space and time twist around each other and together, form the very fabric of nature. We’re quite used to think that time is something “very special” and we have such a stupid thought because we haven’t traveled fast enough (by fast, I mean – closer to the speed of light). Okay, the “\Delta” allover the figure indicates the intervals of parameters (just the difference). \Delta s is the spacetime interval, which is an invariant quantity for inertial motion. The space interval is not a big deal (we know that).

But, note the time interval. We’ve multiplied a factor, c with it. Now, the thing makes no difference from a distance vs distance plot. I apologize for stuffing you up with hard words & non-boiled vague ideas (in case you’re a beginner), but I’m sure that you’ll be able to grasp the whole thing in my future posts. Alright, so what’s \Delta s? Pythagoras may well enlighten you with his work,

\Delta s^2=\Delta x^2+(c\Delta t)^2

Thanks for the workaround Pythagoras. But, bad luck..!!! That’s wrong. Like I said, Pythagorean math has to be modified into a new one (in order to maintain the invariant quantity), which goes like this…

\Delta s^2=c^2\Delta t^2-\Delta x^2

“From where did that “minus” sign jump in?” is something I should preserve for future posts (things are queuing up for the future, prepare yourselves!!). But, this new transformation produces all those miserable effects such as length contraction, time dilation, mass-energy equivalence, etc. especially with the hooked up c in it.

Why was light “chosen”?

In the 19th century, Maxwell did the most profound unification by amalgamating electricity, magnetism and light which were previously thought of as entirely different things. While we don’t need much history for today, we should know that a constant came out of his magical set of equations. The constant was the speed of electromagnetic waves.

c= \frac{1}{\sqrt{\mu_0\epsilon_0}}

At the start of 20th century, Einstein came forward with a postulate that the laws of physics are the same in all inertial reference frames. This keyword “inertial” is very necessary. All along the way, we were talking about inertial observers and inertial motion. So, what’s this “inertial”?

If the motion is uniform, then it’s inertial. Frankly, there shouldn’t be any acceleration. Or, you can also put it this way. Inertial frames are where the Newton’s laws of motion work. In other words, you throw a ball at rest or at constant motion, the way it falls, is still the same. But, if you were to accelerate forward, the ball would go backwards, or if you were to spin around (when your acceleration is towards the center), the ball would start going crazy (as Coriolis comes into play). The latter things are non-inertial. Even more simply, in an inertial frame of reference, you can’t do any physical experiment to conclude whether an object is in motion, or not (whereas in an accelerated frame, you can). A simple experiment would be a kick. When the bus starts suddenly, you experience a jerk, by which you’re thrown back. By that kick, you’ll be able to conclude that it’s you, who’s in motion.

Hence, Maxwell’s magical equations should also be invariant for inertial motion, which explicitly says that the speed of light (electromagnetic waves) is a constant for all inertial reference frames. That’s it, there you go (two postulates of the fabulous special relativity)…

So, the physicists can rely on this thing, being an universal constant. There’s nothing wrong in choosing it as a factor. Or, is there any?

What light year suggests?

Years went on, and a lot of theories and further experiments confirmed the predictions made by the special theory of relativity, especially the fact that light is the fastest speed at which any kind of information can travel. How it became the cosmic speed limit, will be understood in the future, when we’ll address how “absolute time” got blown to hell, how mass unites with energy, how exactly these weird things such as length contraction and time dilation happen, etc.

What happens when the both the parameters in the spacetime interval equals each other? \Delta s becomes zero. This means that the events occur simultaneously. In reality, no massive object can experience that, unless the objects are situated at the same position at the same time, or it’s a photon (or any other massless particle, say gluon, or graviton which always cling to the speed of light). So, there’s always a delay in any other case. 3 meters? The delay is 10 nanoseconds. Moon? The delay is 1.26 seconds. Nearest star? That would be a 4.24 tough years..!!! Looks like it’s true that when you’re far apart, you can observe the event’s past. And, it is a fact…

More simply, if you were an astronaut, having high tactical gears & stuff, surfing near Gliese 876d with a telescope and look at Earth, you can observe me, preparing to visit preschool for the first time. That’s enough fun. Here’s some food for your thought (and to confuse you much, assuming you’re a beginner).

I plagiarized this from Stephen Hawking’s “A Brief History of Time“. These are the couple of things to be noted in this spacetime diagram. Recall that every single phenomena happening in the universe can be plotted in the diagram to represent the “spacetime event”.

  • The time axis is plotted in light-minutes.
  • Recall that information can always travel only at c.

Don’t wonder by seeing “light cone”. It’s nothing but the area which the light covers in the spacetime diagram. You see a triangle because, the thing is 1D space and 1D time. Would the space be 2D, you’d be able to see the beautiful 3D coned shape. What you see in the above image, is a catastrophe. The sun dies. But, these civilized people at Earth will come to know about that, only after 8 minutes…

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