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It is defined as $x\cdot x = 0$. How is this justified?

Also, why does the backward light cone need to exist? It is said that light emitted in a backward cone would pass through the origin. But why? Wouldn't it just create another forward cone with a different origin?

And the most important thing is: what is this Light Cone concept actually about? I don't see how this concept came about.

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closed as unclear what you're asking by WillO, Jon Custer, John Rennie, Michael Seifert, Johannes Dec 13 '16 at 23:07

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ Have you read the wikipedia light-cone page? What about Prof. Norton's description? $\endgroup$ – DilithiumMatrix Dec 12 '16 at 3:52
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    $\begingroup$ The corresponding Wikipedia article seems pretty good at explaining the siginificance to me. Sample: "[...]the light cone plays an essential role in defining the concept of causality" Can you be more specific about what you want to know? $\endgroup$ – ACuriousMind Dec 12 '16 at 3:52
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    $\begingroup$ Re your first question: Even numbers are defined to be multiples of 2. How is this justified? A zoo is defined to be a place where wild animals are collected for study, conservation, or display. How is this justified? What does it mean to "justify" a definition? $\endgroup$ – WillO Dec 12 '16 at 4:33
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    $\begingroup$ @sequence we are talking about points in spacetime, there is no "sooner or later" because the time is specified as well as the position. It's purely a statement of "if information came to the event at the origin from any point outside of the backwards light cone, or information goes from the event at the origin to any point outside of the forwards light cone, then it has traveled faster than light." (That's why it's the light cone.) $\endgroup$ – CR Drost Dec 12 '16 at 5:45
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    $\begingroup$ The $x \cdot x = 0$ definition is a shorthand for $c^2 t^2 - x^2 - y^2 - z^2 = 0$ which says $c^2 t^2 = r^2$ in spherical coordinates which says $r = c~|t|,$ it describes a sphere of light expanding with speed $c$. $\endgroup$ – CR Drost Dec 12 '16 at 5:50
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From what I understand, for an observer traveling at a constant speed, the light cone is the surface such that an observer at the origin (apex of the cone) will not be able to reach a future event (the future light cone) if the event occurs on the outside of the future light cone. Similarly, an observer at the origin would not have been able to cause or witness an event outside of the past light cone. This is due to the fact that, according to relativity, nothing can travel faster than the speed of light. Because of this, the light cone is given by the equation, in 3 dimensions (time being on the z axis) as: $$t = \frac{1}{c}\sqrt{x^2+y^2}$$ To simplify, let's take y=0, which reduces this equation to ct=x (which is the light line in two dimensions). When the derivative of both sides is taken with respect to time, we observe the velocity along the light line: $$\frac{dx}{dt}=c$$Since nothing can travel faster than the speed of light, nothing can (traveling at a constant speed) travel outside of the light line. This can once again be generalized to three dimensions and the same can be said about the light cone.

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If a vector between two events $A$ and $B$ lies inside the light cone, then the time ordering of $A$ and $B$ cannot be changed by any boost. Thus, even though simulteneity is relative, we can still uphold a notion of causality in special relativity, owing to this order invariance. The time interval between $A$ and $B$ can of course change between observers, but the order cannot. If the vector lies outside the light cone, then one can always find a boost that reverse the time ordering of $A$ and $B$.

Given this wholly geometric fact of time order invariance for within-lightcone vectors, we can now make some physical observations and reasonable postulates. For example, suppose I cook boiled eggs for breakfast, and my fridge and kitchen is, say, ten light minutes away. Suppose I raise a flag (event $A$) when I decide to have boiled eggs for breakfast, and then event $B$ is my sitting down to eat them. Leaving aside the time needed to boil them (let's assume we have some futuristic cooking process that takes vanishingly small time), then if I could travel to my fridge at greater than the speed of light, some observers would see me eating my cooked eggs before I decided to eat them! Thus it seems reasonable, if we postulate that the orders of irreversible natural processes are not to be observer dependent, that travel at greater than $c$ is impossible.

Contrapositively, it is this geometric fact of the lightcone and the invariance of time direction of all vectors within it that even allows a notion of causality in special relativity, as long as we postulate that causally linked events can only be linked by within-lightcone vectors. It can be shown on very general symmetry grounds that the Lorentz transformation is linear and homogeneous and, if the transformation depends continuously on relative velocity, then this means that the only possible transformations mixing space and time are rotations or boosts. But rotations are ruled out if we want to uphold a notion of causality as discussed above, because one can always find a rotation to reverse the order of events. The lack of null vectors and light cones thus rules rotations out, so the transformation must be the signatured Lorentz transformation, with boosts rather than rotations.

This is what ACuriousMind, and the Wiki page, mean by the quote, "*[...]the light cone plays an essential role in defining the concept of causality" *".

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There are a couple things to understand about light cones. Your $x$ vector is made up of 1 time component and 3 spatial components. However, when you take the dot product, you actually get

$$x \cdot x = -x_0\cdot x_0 + x_1 \cdot x_1 + x_2\cdot x_2 + x_3 \cdot x_3$$

Notice the negative sign in the first term. This arises because of the way dot-products are defined in Minkowski space (where the metric tensor looks like an identity matrix with the top-left term -1). Your question is "how is this equation justified."

  • If the dot-product is positive:
    • then it means that two events are separated by space. That they happen without being aware of the other event.
    • example: You post a question on Stack Exchange. At this exact same moment, a star goes supernova 10 million lightyears away. There is no causal relationship
    • The vector lies outside of the light cone
  • If the dot-product is negative,
    • then the two events are separated in time. One event influences the other event
    • example: The sun goes supernova. Eight minutes later, the Stack Exchange servers melt under intense heat. There is a strong causal relationship.
    • the vector lies within the light cone.

Any point within a light cone can causally affect (if below the origin) or be affected by (if above the origin) the event that the origin represents. One doesn't "emit" light backwards--rather the origin is sensitive to light that was emitted at a previous time.

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