light cone

A light cone is the cone extending in timespace in front and behind of any event. The boundary of the light cone is the boundary of events that could have possibly influenced the event in question. For example, an event which is a light year away from a test location couldn't influence events at that location for a year. (Ya, I know it's more complex than that -- see a Relativity text for the full story.) The forward light cone is all events that the given event can influence.

Given a metric g_ab, the formal definition of a light cone based as some point x^c is

LC(x^c) = {y^c|g_ab(y^a-x^a)(y^b-x^b) = 0}

part 4


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My bourbon slushie is nothing but a small brown puddle cradled in the bottom corner of my plastic cup. I'm staring at the liquid, listening to Zoe as she pushes a couple more pictures under my nose.

A scrap book. Polies all go to Outward Bound before they winter over. Someone decided a ropes course and physical challenges would would initiate strong ties in a team of disparate individuals. Some psychologist decided it was important for these ties to begin before deployment. Odd birds could be weeded out. Incompatibilities surfaced. Everything ironed out before the ice.

"But they do very little about anything and there's a huge flaw. One bleedingly, fuckingly obvious thing," Kyle says, walking by. Zoe shoots him a look. Occasionally each of the other polies stare at me as if sizing me up. I'd introduced myself to all of them earlier. Seems like they're waiting for me to explode.

"I am not 'The Thing'," I say, and everyone laughs.

It's not anyone's fault.

Why did I think that?

"Here's Jana," Zoe holds the photo to my face to get my attention. "Isn't it amazing? We got her at the apex of flight."

The picture is of a sheer rock wall. Two climbers are framed in the shot wearing harnesses festooned with bundles of carabiners and cams. One hangs from a purple rope, arm outstretched, the other, Jana, is weightless, falling finger tips touching the hand of the first.

"It was the part of the climb where you had to jump to this little ledge. She was afraid to go. We tried for hours. Nobody could make her try except Greg. He talked her in. She said it made her..."

Her voice is hypnotic and I'm half drunk. Blue and Red LEDs flash across the empty makeshift dance floor. Across the room, Jim and Kris lay entwined in a face-to-face lock that hasn't ended from the moment we arrived. No one seems to care, but it's distracting me. How were we supposed to do whatever we were supposed to do with them having sex over there?

"...Greg..." It's Zoey.

"Greg." She puts a finger under my chin and I let her turn my face toward her.

"Glenn," I say. "Two 'n's."

"Glenn. Sorry. I was thinking about Greg. Did you know--?"

The door to the lab opens and two polies come in, brushing ice from their faces and their green parkas.

The taller of the two says, "Incoming. Definitely."

Someone kills the music. Hits the fluorescents that drown us in blue white daytime so we have to squint. In the bright light, everything is sharp and ugly and covered in dirt.

"How many?" comes a voice I haven't heard before.

"Two. We have about five minutes."

"What's incoming?" I ask. No answer.

Zoe packs up the photos and looks at me. I notice her eyes are bloodshot. Too much drink. Too much.

"You could help," she says.

With what? "You only have a couple of pictures--?" I say, knowing that can't be it.

Kyle bolts from a sofa. Bushy black Jerry Garcia hair and beard. "I want to get out of here. I gotta get out of here. I want to go home."

The guys in the parkas are on him. One in front, hands on his shoulders, trying to calm him down. One in back, looks to me like he's patting him down.

I hold out my hands palms up. "What? What the hell's incoming?"

Now I can see the names on the parkas. Rob. Harlan.

Rob says, "You tell me. They're yours."

"Do we go back to station, or stay here?" Harlan asks Rob, and I'm trying to figure out what they mean. Everyone is up and doing something.

"Power's out at station," says Harlan.

"No it's not. Was up two hours ago," Jim says. He and Kris pull on their ECWs. "Pumps are still down. No sound in the pipes."

Rob says, "Ok. Everyone suit up, just in case. I don't want anyone in the tunnels in case of collapse. And nobody, go outside."

Kyle pushes past Rob and Harlan. Faces off to me. "I want to go home."

"We're at pole," I say, trying to figure out how I can help make him understand we are home. There's no getting out for four more months.

To Jim: "What's incoming? What's happening?"

Kyle says, "It's your fucking nightmare."

"Cut it!" Jim spins Kyle around. "Stop. Keep your mouth shut."

Something thumps the ground, low and deep. The building sways like a boat on the swells. A bottle of Jack Daniel's falls off a table.

Kris says she's got it, and starts mopping up.

"Jim?" I'm asking, praying it's an ice quake. Happens all the time. Ice shifts, mini earthquake.

Building sways. Books come off shelves. Polies, suddenly sober, cap their booze and pack the bottles in boxes on the floor. The streamers sway.

"Jim?"

He ignores me. Rob's gone out. "How far out?" Jim asks Harlan.

"By now, not far."

"Me or him?" Jim asks.

Harlan looks at me. Jim looks at me. Everyone looks at me.

"Hey, guys--" I say. I'm begging for an explanation no one will give.

Jim holds out my parka. "Go."

"Jim. What the fuck?" I say. And I'm putting on my ECWs without knowing why. The ground shakes again. Low thuds transmit through the buildings structure like war drums from an advancing army.

He pulls the pistol from my pocket and hands it to me when my gloves are on.

"How many rounds in a Browning HiPower?"

"I dunno," I say. I never knew.

"Yes you do. Think."

I say the first thing that comes into my head. "Eleven. Ten in the clip, one in the chamber." How the hell do I know that?

"Where do you have to hit them to kill them?"

"Kill what?" I say.

Zoey takes my arm, looks into my eyes. "Greg? What's coming? What did you bring?"

"What the fuck's with you people? My name is Glenn."

The thudding is louder. Something hits the building. It sounds like a cable. Rope.

Rob bursts in. "They're here. What's the plan?"

Jim says, "We're sending him out."

The hell they are. "I'm not going anywhere."

This time, whatever hits the building shakes the floor and polies fall. I brace myself against the wall. Through the open lab door two bright shapes pass my feet. They curl up around Zoe's feet.

What am I seeing?

Tiny glowing dinosaurs?

"They're scared of what's outside," Zoe says.

"What's out there?" Jim says to me.

"I'm just here to fix the radios," I say.

"Listen to me, asshole," Jim grabs my arm in one hand and my face in the other. Eye to eye. Nose to nose. "You're going to get us all killed. Your pussy-ass crying is gonna waste every living soul on this continent. Those things belong to you. Get a backbone or we're all dead."

I want to cry. My guts are in knots. "I don't know what's happening."

"Yes you do."

"Stop it. Please. Tell me what's happening."

"Get out there and kill those things."

"What things? What is wrong with you people?"

"Spiders," Rob says. "That's what they look like to me."

Dear God. What? What? I can't get the words out.

"Eleven shots. You're bound to hit one, at least," Jim says.

"Jana needs you," Zoe says.

Kyle shouts when the building rocks again. He falls to his knees clutching his head in his arms screaming, "Jana's dead. You killed her. Now we're all dead."

Tapping on the walls. Something's climbing the wall. Up from the ice. Onto the roof. Dust and tiles fall from the ceiling.

Nightmare. Nightmare. "This is not real." I will not let this be real.

I grab the Browning from Jim's grasp. Push the safety off with my thumb.

Out in the hallway. Aim up. Pull the trigger.

The concussion is deafening. My ears ringing. Polies shouting behind me. Don't shoot holes in the building.

"What is wrong with you fucking people?" I scream over the ringing in my ears.

Through the door. Into the blast of nothingness. The ice that burns and kills what it touches. Something dark ahead. The size of a truck. The size of a bulldozer. Size of a building. Size of death.

Fire into it. The Browning's not so loud outside, but I can't hear anymore anyway. Something bright in the distance. Tall aluminum tower. Skeleton of silver. Ring of light. The comms towers. People standing. Flashlights aiming high so the beams make a cone.

Nothing. The black thing is inert. It wasn't moving anyway. I am not afraid of you. I am not afraid of your freaking death. I am not afraid to die.

Something dark streaks across my face. Cuts into my cheek. Sharp. Burns. Behind me.

Turn into the blackness. I fire the gun until it stops pulsing in my hand.

The darkness falls away so I see the building again. I am not afraid of you. I will die for her in a heartbeat. Come on. Kill me.

The wind bites me where my balaclava pulled away. I am not afraid of you.

"Kill me, you son of a bitch. You got nothing better than that?" Who am I yelling at? "You gotta do better than that. You just scratched me."

Who are those people?

Off in the distance, beyond the dead spiders, next to the communication tower, a circle of people, red and green parkas, flashlights aimed toward the top of the tower.

Who are they?

Jim steers me back into the lab in the clean air building. Takes the gun out of my hand. Sits me on a spot in the sofa the polies make clear. It's warm and I can't even tell.

"Did good. You got 'um," Jim says. "It's over." His words suck the energy from me. Whatever was in me is burned out. Cinders. The rising smoke is my dead soul.

I manage to say, "They're tropical. Would have frozen to death, anyway. When I was young we lived in Florida. That kind of spider..."

"Don't think about it," Zoe says. She takes off what's left of my balaclava, and taps the tear in my cheek with a piece of gauze that comes away bloody.

"Yeah, just don't think anymore," Kyle says. "I want to go home."

"They're not twenty feet tall back in Miami," Jim says, smiling.

Rob says, "I'll get the D5 out and bury them like the others as soon as we can get back to the garage."

And I think of the people around the comms tower. The christmas tree of flashlight beams pointed toward the tower's peak. "What's on the comms tower? Who are those people?"

"In time," Zoe says. "Jana's asking for you." She sticks a piece of gauze to my face with white surgical tape.

"Jana..." I'm thinking of her now. Long brown hair tickling my face. Bright blue eyes. That smile when she says, "too late now."

"Too late now," she says to me in my mind. We're here. If you didn't want to have sex, we shouldn't have wound up naked in bed.

"That's not what I was planning."

"I have plans, too," she says. She says. She looks at me like everything she needs comes through my eyes. All I need comes from the warmth of her. Her skin a caress.

Lord, how I miss her. I don't want to live. "How did I live?"

"Shush," says Zoe. "Shush. You've had a hard day."

My eyes are closed and I'm with Jana. I open them, I'm back in the hurt.

When Zoe is finished patching me, she says, "Jana's in medical. Why don't you go see her?"

Yes. Why don't I? I just killed two giant spiders for her. They didn't kill me, so I can see her.

I need to see her.

I forgot this. Somehow.

I promised I would never forget, and I did.

That I love her. This falling is love.

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next episode: amanda speaks
first episode: stendec

In special relativity the spatial distance between two events and the time between when they occur become things that depend on which observer you ask. Observers in different frames of reference will record different positions and times for events. So if the spacing and timing of events is no longer an objective fact, we have to ask what happens to our concepts of events happening in the future, past, or elsewhere (meaning in another place). The light cone is a concept that helps us divide up spacetime and understand what we can say about events in relativity.

We'll start this discussion by drawing a spacetime diagram, so you might want to check out that node first if you don't know about them yet. The light cone is drawn by choosing an event, which we usually just put at the center of our diagram, and then drawing two worldlines. One is a line that corresponds to a light ray moving in the positive x-direction, and one a light ray moving in the negative x-direction. So, each line should have a slope of 1, representing that it is moving with the speed of light in vacuum. I've done the best I can with ASCII art, so pretend those are straight lines with a slope of one. Here, then is the standard spacetime diagram for the light cone.

  v/c=1           ct ^                v/c=1
     \_              |              _/
       \_          future         _/
         \_          |          _/
           \_        |        _/
             \_      |      _/
               \_    |    _/
   elsewhere     \_  |  _/      elsewhere
                   \ | /
  -------------------+-------------------> x
                  _/ | \_
                _/   |   \_
              _/     |     \_
            _/       |       \_
          _/         |         \_
        _/           |           \_
      _/           past            \_
     /               |               \
                     V

The two lines we've drawn make up the boundary of the light cone. It's called a light cone because it's made up by two lines representing light beams and because it draws out two cones (technically one, two naped cone). They'd be more like actual cones if we added another space dimension coming directly out of the monitor at you, in which case the set of all lines coming out of the origin with v=c would make a cone¹.

Dividing Up Spacetime

The light cone divides up spacetime around the event in question into past, future, and "elsewhere". So let's talk about what that means. The important part to understanding these differentiations is the spacetime interval (sometimes also called the spacetime invariant). From the fact that all observers in special relativity measure the same speed of light one can argue that the the quantity

Δx² - c²Δt²

is the same constant for all observers. Meaning that though the distance in space Δx and the length of time Δt between two events may be different for two different observers, they will both agree on the same value of the spacetime interval Δx² - c²Δt². Based on that fact we can start to reclaim some idea of past, future, and elsewhere.

Past and Future -- Timelike Separation

A point inside the light cone has |x| < c|t|. That means that x² - c²t² is negative. However, we've said that this is a number all observers can agree on, so that means that it's negative in all frames of reference. For the total to come out negative, we know that t' (the time for that event in some other frame of reference) can never be zero. If we know that t' never crosses zero, then it seems safe to assume² that if t starts out greater than zero, it is greater than zero in all frames of reference, and if it's less than zero it's less than zero in all frames of reference; thus, we've justified our labels above. Something inside the future portion of the light cone happens after our event for all observers, so we can safely say it's in the future. An event in the past portion of the light cone happens before our event in every frame of reference, so we can safely say it's in the past. For a discussion about time ordering in relativity in more depth, see my writeup on relative simultaneity. Now, it should be noted that we can't say too much about whether an event inside the light cone happens at the same place in space as the origin or somewhere else. If all we need is x² - c²t² to be a negative value, we can find a frame of reference where x'= 0 and -c²t'² is equal to the required constant. So for events inside the light cone there is always a frame of reference where the two events are coincident, meaning they happen at the same place. When one event falls within the light cone of another, and, thus, they have unambiguous order in time, the events are said to have a timelike separation.

Elsewhere -- Spacelike Separation

For a point outside the light cone |x| > c|t| and x² - c²t² is equal to a positive number. Now we can play the same trick and say that, since this must be the same positive number for all observers, x' cannot be zero in any frame of reference, since then x'² - c²t'² couldn't come out positive. We could then infer again that x doesn't cross zero, which is true with only one dimension of space but is not true with the three dimensions of space we actually have, so we won't focus on that³. Still, the fact that x' will never be zero in any frame of reference means that we can unambiguously say it is elsewhere than at the origin. We can't, however, say to much about when it is, because it is always possible to find a frame of reference in which t'=0 and x'² has the positive value of the spacetime interval. When one event falls outside the light cone of the other, and the two events unambiguously happen in different places, they are said to have a spacelike separation.

On the Light Cone -- Lightlike Separation

There is one more class of points that we haven't discussed, and those are the points that actually fall on the light cone and have |x| = c|t|. Because the speed of light is the same for all observers, the worldlines that make up the light cone will be the same ones in every frame of reference, so a point that is on the light cone in one frame of reference is on the light cone in every frame of reference. Going back to the spacetime invariant, we can see that x² - c²t² = 0 for all the points on the light cone. If one event falls on the light cone of another, they are said to have a lightlike or null separation.

Though it is not a postulate of special relativity, it is generally accepted⁴ that no information or influence can propagate faster than the speed of light. This gives our definitions of past and future even more meaning. Since only events inside the future portion of the light cone can be connected to the origin by a worldline with a speed less than light, an event can only influence events that fall inside or on the boundary of the future portion of its light cone. For the same reasons only events in the past portion or the boundary of the light cone could have influenced the event at the origin. Thus, only events inside or on the boundary of the light cone are causally connected to the event at the origin. Furthermore, if the origin marks the position of a body moving though space, then it can only possibly move into the future region of the light cone and can only have come from the past region.

Light Cones in General Relativity

For completeness, I should also mention that the concept of a light cone also comes up in general relativity. In GR the light cone is a local concept that makes sense when you're talking about spacetime close to a particular event, since on a small scale spacetime looks fairly flat. The idea is basically the same, except that the spacetime interval that we've been using is replaced by distance according to the particular metric of that curved spacetime. The light cone is then the set of trajectories that give zero distance from the event in question, just as we've done above, and again it divides spacetime locally around that event into past, future, and elsewhere. One example in GR where light cones are often used is in the Schwarzschild geometry around a black hole. The light cones of events far away from the black hole look pretty much like the ones for Minkowski space that we analyzed above, but near the event horizon the light cones begin to tilt with the future light cone tilting more inward toward the black hole. Finally, at the event horizon the light cones are tilted so much that the entire future portion lies inside the event horizon; thus, an object at the event horizon can only go into the black hole as it does forward in time.

1 Imagine taking the picture above and rotating it around the time axis.

2 This follows from the fact that transformations between different frames of reference are continuous. This would generally be assumed from the outset, and if we know about the Lorentz transformations then we can readily show that it is so.

3 In more that one dimension of space we can easily make x' = -x just by choosing a frame of reference that is rotated 180 degrees around one of the other axes. To look at it another way, there are clearly many ways to go from x to -x without going through the origin in 3 dimensions. On the other hand, for events with timelike separation they are confined to stay inside the light cone, so they'd have to go through the origin.

4 It is accepted for good reason. Particles or information traveling faster than light introduces paradoxes into the theory and erodes the entire idea of causality. More importantly, no such phenomenon has ever been observed.

Note: The light cone is apparently also sometimes called an event cone or a null cone.

Sources: My own knowledge of special relativity.

Though I didn't really consult any sources to write this, if you want to look at some, here are some suggestions:

The book I originally learned from (not necessarily recommended)

A. P. French, Special Relativity

A well respected introduction to special relativity from the view of geometry (spacetime diagrams) at an introductory/intermediate college level

Taylor and Wheeler, Spacetime Physics

Finally, most general physics text books will have a chapter somewhere near the back about relativity, which generally will contain a section on this subject.