Light Cone

bulb on a stringSo imagine that you’re standing at some spot, holding a big round bulb that can shine in all directions. At a particular time, you switch on the bulb and it shines out brightly. Who has a chance to see the light from your bulb? Let’s not worry about things like buildings, cars, etc., getting in the way of the light, but just imagine it can shine free in all directions. You’ll realize that there is a whole ball of light expanding away from the bulb – light going away from you in all directions. (The edge of that ball describes what is called a sphere, a two-sphere to be precise, denoted [tex]S^2[/tex] in the trade.) The sphere expands at the speed of light. You might imagine that eventually every point in space around you would be reached by your expanding sphere (let’s stay local and not worry about the expanding universe and so forth), and you’d be right. But when? When does it get to some particular point that you might care about? As soon as it takes the light the time it takes to get there, of course. It can’t get there any sooner than that.

Let me put it another way. Your sending out the light is a particular “event”. It took place at a particular place -where you stood- and at the particular time you flicked the switch -three o’clock, saya. One second later, since light travels at about 300,000 kilometers per secondb, all points about 300,000 kilometers away (in any direction) will receive that light. A person that distance away looking your way at that time (1 second later) will see the flash of light from you. Any time less than a second before, they will not have seen that light. It could not have got there yet. No signal from you could have got to them faster. You cannot have any effect on that person sooner than one second after you emitted the light. You can “cause” nothing to take place at their position, earlier than one second later. We can call that position (300,000 kilometers away) and such a time (half a second after 3 o’clock) an “event” as well. So said more succinctly, the event of you sending out the light is not “causally connected” to that event (300,000 Km away, half a one second later), or similar events, such as 1 second later but 400,000 km away.

expanding circlesIn physics, it is very important to appreciate the above, and people spend a lot of time drawing pictures to make clear what is going on. Let’s try to do that. Rather than three spatial dimensions, people often consider two of them, just for ease of drawing what we’re about to draw. In this case -imagine the two dimensional world is the surface of your desk or dining room table- the ball of light that is expanding away at the speed of light is a disc, and the edge is a circle (we’d call that [tex]S^1[/tex] if we were trying to impress our friends in the bar. We’ll stick to the word “circle” here). Everything I said above is still true, I’ve just replaced a sphere with a circle.

lifting the circlesNow for a useful diagram. It will explicitly represent what is going on in time as well as space, and it actually called a “space-time diagram”. Let us call three o’clock “time [tex]t=0[/tex]”. A tiny fraction of a second after that time, the circle of light expanding away from you is quite small. A bit later, it is bigger, as the light is moving away from you. Later, a bit bigger… One second later, the radius of the circle is 300,000 km, two seconds later it is 600,000 km, and so forth. So we have a family of growing circles – the later the time, the bigger the circle. We can draw all these circles on the plane, all centered at the place from where the light was emitted. Just for fun, let’s lift the circles off the table. We raise a circle higher off the table according to the later elapsed time that it represents. What we are doing is representing the time that has elapsed in terms of how far we lift them above the table, positioning them along a line in a vertical direction in our diagram (we just chose vertical for no good reason… it does not have anything to do with vertical in space). If we are careful to accurately preserve the proportions to represent accurately the time elapsed for each circle, we’ll get a familiar shape.

lightcone sketchYou know this shape. It has a name. It is a cone. The vertex of the cone is the original event of you (or your two dimensional avatar) hitting the switch to the light bulb. Any point in this diagram is also an event, since it represents a place (in the two dimensions) and a time (how far in the vertical). Any event on or inside the cone (and you must imagine it going on forever), where the ice-cream would be, can be reached or affected by you (you’re at the apex, recall) since you don’t have to go faster than light to get to it. Any event outside the cone cannot be affected by you – it is not “causally connected” to you – since you’d have to send a signal or other effect at faster than the speed of light.

You now are acquainted with the “light cone”, the central concept in Einstein’s Special Relativity.

full lightcone sketchYou can go further and extend the cone downward in the other direction…. into the past. What does that mean? Well, it is talking about the events from the past that could possibly effected you. A point on the cone at some earlier time than when you emitted the light (say, one second, 300,000km away) is causally connected to you since someone 300,000km away could have flashed a light bulb, and you’d receive the light at the original event (your original spot, at 3 o’clock – they emitted it at 2:59:59, and it took one second to get to you, you see.) An event one second earlier than [tex]t=0[/tex] but say, 400,000 km away, is outside the cone. It is again causally not connected to you. They are too far away for emitted light to have reached you one second later – it’s still 100,000 km away at that time.

So the top part is the “future light cone” – it separates points representing events that you can effect from ones that you can’t, and the bottom part is the “past light cone” – it separates events that can affect you from events that cannot. Light cones are useful for lots of reasoning. Maybe I’ll talk more about them some other time.

Well, that’s it. In a sense, you actually carry a light cone with you where ever you go, representing the borderline between the possible and the impossible, the accessible andthe inaccessible. Look after your light cone. Here are some solid ones you can buy and put on your mantelpiecec:

lightcone by tamsin van essen

-cvj

___________________________________________________________________________________
aA time and a place is an event. Recall that party you had in your room or apartment last night? That was an event, right? It had a place and a time, like you put in the evite, the text, or in embossed faux gold lettering on the invitations you sent out. You need both a time and a place to specify the event. Your apartment is not a party, or any other type of event. 10:00pm is not a party. Your apartment at 10:00pm, however…. Absolutely.

bIt’s actually 299,792.458 kilometres per second.

cThis is a picture of some of the lovely work of Tamsin van Essen. You can see more here. My writing more about subjects touched on by some of the her pieces, and you reading it, may well be inside all of our future lightcones.

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28 Responses to Light Cone

  1. Romain says:

    The post is interesting but Einstein already explained me this concept in one of his books. I actually prefered his more direct introduction, probably because I study scientific subjects, so I am used to this kind of reasoning.

    But that’s not the point of my response. I don’t how easy it is for you, but it would be very handy if you could link the small letter indicating a note to the note, and put a similar link backward. When the article is as long as this one, it is not very easy to scroll down to the note and come back to the text.

    That’s my feature request 🙂

  2. Neil says:

    At this website, you can generate an RSS feed of your personal light cone (starting from your date of birth), which notifies you when it passes a stellar object. Quite cool!

  3. Plato says:

    I certainly appreciate the lesson on light cones.

    Also, ways in which one may of looked at “how light cones may be used?”

  4. Carl Brannen says:

    So if the event is a point where a quantum mechanical measurement is to be / was made, should the event be represented by a wave or a particle? Does it matter if the event is discussed “before” or “after” the measurement is made?

  5. Bee says:

    Ever noticed the ‘X’ as favicon on my blog? It’s not an X – it’s a lightcone 🙂 Very nice post!

  6. So how about them quatum entagled objects? Are they exempt from the light cone?

    So how about them stuff at the edge of the Universe? Is your flashlight beam ever gonna make it there?

    299,792.458 kilometres per second. It’s not just a good idea – it’s the law!

    Having done an experiment to measure the speed of light, i have to say that getting more than two significant digits is pretty impressive. We were within 3%. Any idea how nine digits were achieved?

    So, at 370,000 years after the Big Bang, the Universe became transparent, and the CMB (which wasn’t yet stretched to microwaves) was sent everywhere at the speed of light. We can see this radiation. That suggests (to me) that either the Universe was bigger than 370,000×2 light years across at that time, or our average speed from that point was faster than light, or the Universe has wraparound topology. Did I miss something?

  7. Aaron F. says:

    Your apartment is not a party, or any other type of event.

    Maybe your apartment is not a party… 😉

  8. Pioneer1 says:

    Thank your for this post. I realized I have not thought about this for a long time.

    I have a question. I think that the fundamental assumption of the light cone concept is that the speed of light is independent of the speed of the projector. I was wondering if this assumption was ever verified by an experiment?

    Thanks

  9. Clifford says:

    I did not use it in the post, but you’re right, it is important. The fact that the structure I described above remains intact for all observers I can get to by a translation, rotation, or boost to a different velocity is the statement of this fact. The entire framework of Special Relativity is built on this. Experimental verification? Too numerous to mention. It’s just about everywhere: It starts with the Michelson-Morley experiment and goes from there. It is built into everything we do with electricity and magnetism, sdo much of what we observe in astrophysics and cosmology, oodles of data from particle physics experiments, and even manifestly into our technology as we comunicate with out satellites, and even use those satellites to locate where we are on planet earth with GPS.

    Cheers,

    -cvj

  10. Clifford says:

    Stephen Uitti:- Entanglement still does not permit you to communicate signals faster than light. So entangled states are not exempt. As to your other question: You’re assuming that the universe is a bunch of matter expanding in a fixed space. This is not the case: The space itself is expanding. See the link that Bee pointed you too.

    Carl Brannen: I don’t really understand your question very well. Events are still events (as defined in the post) in Quantum Mechanics or Quantum field Theory.

    Cheers,

    -cvj

  11. Pioneer1 says:

    Thanks for your reply. One more question. At some point you simplified to two dimensions and obtained the light cone from that simplification but never went back to spherical. I am not sure and I have to think about this more, but I believe there may be a difference. For instance, if you continue with spherical propagation the lower part of the cone as “past” does not make sense since sphere has different symmetry. What am I missing?

  12. Clifford says:

    No, that’s not correct. The simplification was just to allow me to draw a picture. Going back to 3+1 dimensions just adds a spatial dimension back, and the circle is a sphere. The lightcone in any number of dimensions is just as simple. You mentioned symmetry. Indeed, it is more elegant to think about this all using the Lorentz transformations, which preserve the structure of the lightcone.

    Cheers,

    -cvj

  13. Some facts about light-cone boundary of 4-D Minkowski space which for some reason have not catched attention of physicists.

    Light-cone boundary, the moment of big bang at the limit of empty Robertson-Walker cosmology, is topologically 3-D but metrically 2-D and therefore allows an extension of 2-D conformal symmetries. Decomposing light-cone as S^2xR_+, where R_+ represents light-like radial direction, you get conformal symmetries of 2-sphere plus new kind of symmetry mathematically very analogous to conformal symmetry associated with the light-like radial coordinate. The isometries extend to infinite-D group: conformal transformations with conformal scaling compensated by suitable radial scaling.

    These magic properties select 4-D Minkowski space to a a completely unique position since they provide a way of circumventing the no go theorems justifying the belief that strings are fundamental objects. I am still wondering is it really possible that despite my considerable efforts I have not been able to communicate these simple facts to them or do they really refuse to take these simple facts seriously.

    More generally, light-like 3-surfaces of Minkowski space or more general spaces of form H= M^4xS, share the same properties and Kac-Moody symmetries can be understood as local variants of isometries respecting the light-likeness property. In TGD the like 3-surfaces of M^4xCP_2 are indeed fundamental dynamical objects and a huge generalization of string model conformal symmetries is implied together with the explanation for the 4-dimensionality of space-time and of standard model quantum numbers. These generalized super-conformal symmetries are implied if one accepts quantum gravitational holography in the sense that physical states can be expressed using data at boundaries of light-cone(s actually) and in the sense that light-like boundaries of more general causal horizons of space-time surfaces code for the quantum physics and space-time interior only provides classical correlates for this quantum physics.

    Those theorists willing to challenge the decision that M-theory is the only possible theory of everything are wellcome to my home page at http://www.helsinki.fi/~matpitka/ containing 15 books about TGD and its applications or to my blog at http://matpitka.blogspot.com.

    Matti Pitkanen

  14. Louise says:

    Contrary to popular belief, a constant speed of light is not necessary for Relativity. The Man himself said constant c is “neither a supposition nor a hypothesis about the physical nature of light, but a stipulation which I can make at my free discretion to arrive at a definition of simultaneity.” (The Collected Papers of Albert Einstein, Vol. 6, Princeton U. Press, 1996, p. 439)

  15. Clifford says:

    I refer all readers to an earlier thread where several physicists addressed Louise’s unorthodox suggestions about Relativity and the constancy of the speed of light. The suggestions were not found to hold water. Follow the discussion in the thread of this post, and the links therein.

    -cvj

  16. Pingback: Check Your Light Cone - Asymptotia

  17. Doug says:

    Why not think of light cones as catenoids, which some use to describe de Sitter spacetime?
    [Derek K. Wise, MacDowell-Mansouri gravity and Cartan geometry, page 6 diagram.]
    http://xxx.lanl.gov/abs/gr-qc/0611154

    Does our future really not consist of the intersection of continuous past light cones. The sun light cone is estimated to be at least about 8 minutes old when viewed in every future second. Other light cones are at least light years away.

  18. Pingback: Centauri Dreams » Blog Archive » Philosophia Naturalis #6 Now Online

  19. we can easily imajin and guess all about the future light cone but its really very much difficult to make the physical interpreatation of the past light cones. Past light cone , the things tht can effect … wat we can interpret it ?? and specially in the case of the light cones how can the things effect on the source of light before emmiting the light beam ?

  20. Rich says:

    People who enjoy this post might also like my series on physics, part of which covers special relativity through spacetime diagrams. The relevant articles begin with “Spacetime and Coordinates” and from there explain the relativity of simultaneity, absolute pasts and futures, the relationship between relativity and causality, time dilation and length contraction.

  21. JAG says:

    Hello quantum teleportation!!!
    We can send the photons at 4c down coax cable
    That ought to mess with your cones!!!

  22. Clifford says:

    No. Not at all. This is a common misunderstanding. Quantum teleportation still requires information to be send by regular (light or sublight) channels in order to work. So it does not violate the light cone. Sorry to disappoint.

    -cvj

  23. Michael Breach says:

    Light cones are in fact geometrical boundaries of the universe. We are in a geometrically closed/boarded space relative to time and the spatial dimensions. since nothing limits extra dimensions including time, we are so to speak in a black hole relative to the infinity of possibilities.

    QM is the result of an object hitting the boundary of this spatial enclosure and consequently flattening out so to speak as to paint the entire geometric boundary and being everywhere relative to it, but at certain places with greater certainty. This flattening out is observed as electro magnetic radiation. objects with appreciable mass
    can never reach this boundary due to the limiting laws

    of special reality, and the infinite energy required.

  24. Iam very much impressed and inspired by the pictorial depiction of light cone.It is wonderful academic tour.But Iam still confused to have a “pre-big bang” concept on light-cone since all causal links are cut off.Is there any possibility within our own universe for a super-luminal velocity to tunnel a “space-like” region around the light-cone? How the ring singularity of Kerr’s Black hole will “cause” a causal punches in this space-time warp.?If the photon is mot a point (massless) particle but a “vibroton” (I mean “string”
    theory)then how will this “light-cone” picture be visualized? I will be extremely thankful on this share of thought.

  25. Clifford says:

    HI,

    I’m so sorry but I don’t really understand your questions. All I can think to do is recommend Kip Thorne’d excellent book Black Holes and Time Warps. That might help with some of the ideas you’re trying to develop.

    Best,

    -cvj

  26. evan says:

    first off thanks for the post, as someone just getting into this stuff i found your description the easiest for me to understand. i have one question: if a future light cone represents all that i can affect and a past light cone all that has effected me, does that take into consideration an event which lies in the ‘elsewhere’ region but affects and event in a light cone that then affects my light cone? in other words, can light cones overlap such that a light cone can affect a something through another light cone (can a light cone event indirectly affect another event that is not part of its light cone). is the answer that since nothing can move faster than the speed of light that there can only be direct effects but not indirect effects? thanks.

  27. Clifford says:

    Hi. Yes, lightcones overlap. Draw it on paper and see. Take two points horizontally separated. Then draw future light cones from them. They will overlap. The points are in each other’s “elsewhere”, but light from them can meet. In other words, light leaving a point called “A” on the surface of the sun two minutes ago was in your elsewhere… obviously not in your future and certainly not in your past since you could not have been at the sun two minutes ago without moving faster than the speed of light. But six minutes from now it will certainly be visible to you. So the light cone of A and your lightcone intersected even though you and A are in each others elsewhere. From there is it easy to make a third lightcone of another point be the intermediary.. just place a detector in a convenient place whose job is to re-emit light that hits it. A mirror will do.

    Cheers,

    -cvj