Any FTL signal can be sent back in time

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Posted October 27th, 2008 in Relativity 2. Tags: , , , .

One surprising consequence of Einstein’s special theory of relativity is that any signal traveling faster-than-light (FTL) can be used to send a message to the past. Special relativity divides the entire universe into three distinct regions as seen by any observer:

  • The timelike future contains all points where you could be in the future if you could travel at any speed up to and including the speed of light.
  • The timelike past contains all events that could possibly affect you right now.
  • “Elsewhere” contains all other events. One example of an event in Elsewhere would be the status of the satellites orbiting Mars right now. Their status right now is unknowable on Earth because there’s about a 30 minute light travel time delay. Importantly, FTL communication affects events in Elsewhere.
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Spacetime diagrams are just plots of any object’s position (in one direction only) versus time. I’ll call the object “Bob” and decree that he can only move to the left or right. Conventionally, these diagrams measure time on the vertical axis (which I call the time axis) and Bob’s position on the horizontal axis (which I call the space axis). This means that Bob’s speed in the left or right direction is expressed by the angle formed between the vertical (time) axis and the line he traces out on the diagram as he moves. Higher speeds translate into larger angles away from the vertical axis. Time and space are also conventionally measured in units chosen so that if Bob (somehow) manages to travel at the speed of light, his path would trace out a line that perfectly bisects the angle between the time and space axes. In other words, light beams trace out lines that make an angle of 45˚ with the time axis.

There are some important lines on the spacetime diagram that radiate out from the origin (the point where position=0 and time=0, otherwise known as “Bob’s current position”):

  • Vertical lines represent Bob when his speed is zero- when he’s standing still. He’s not moving in space, but he is moving inexorably forward in time. Thus, the vertical (time) axis points up towards “the future” and is defined by the path traced out by Bob when he isn’t moving.
  • The lines at 45˚ away from the vertical (time) axis represent the path traced out by the light from Bob’s flashlight as he shines it to the left and right.
  • Lines at much smaller angles could represent Bob when he’s running to the left or the right.
  • Lines at angles greater than 45˚ away from the time axis would represent something going faster than the speed of light, such as the proposed FTL signal.
  • A horizontal line pointing along the space axis represents infinitely fast motion, such as the signal from an instantaneous communication device. It’s more useful, however, to think of the space axis as the set of all events in the universe that are occurring right now. In other words, the horizontal (space) axis defines what Bob considers “the present.”

Imagine Bob is standing at the origin of the spacetime diagram. His timelike future on the diagram looks like an inverted triangle. Its sides are physically defined by the path traced out by light (headed to the left or the right) that leaves Bob’s flashlight at time=0. Bob’s timelike past is an upright but otherwise identical triangle. His timelike past is physically defined by the path traced out by the light from distant stars that happened to hit Bob’s eyeballs at time=0.

So far, so good? Good. Here’s where it gets weird.

According to relativity, every observer in the universe has to measure the speed of light (in vacuum) to be exactly the same speed regardless of the observer’s velocity. Although this may seem like an innocuous statement, it’s not hard to come up with a scenario where the consequences of this little rule can blow your mind. For instance, imagine Bob is in a spaceship traveling away from Alice at half the speed of light when she shines a flashlight at him. Alice measures the speed of the light leaving her flashlight to be the speed of light- a sensible result. But consider this: when the light from Alice’s flashlight reaches Bob and he measures its speed, what speed will he measure? Most people (myself included) want to say that Bob’s measured speed will be “missing” his relative velocity- that his measurement will reveal the light to be traveling at half the normal speed of light. This sensible, intuitive answer is simply wrong; he measures the light to be traveling at exactly light speed. How is it possible that both Alice and Bob measure exactly the same speed of light in this situation?

In order to answer this question, it’s important to realize that in special relativity all motion is relative. Bob can draw a spacetime diagrams with axes that center on him, but so can Alice. The diagrams they make will be different, but keep in mind that they’re both correct. Alice and Bob have equally valid claims[1] to being “at rest” once Bob’s spaceship stops accelerating and starts coasting at constant velocity, so neither of the diagrams is “the right one.” This means Alice has to be able to draw a spacetime diagram for herself that explains why she measured the speed of light to be the correct value, but she also has to draw Bob’s spacetime diagram from her perspective in a way that explains why Bob measured the speed of his flashlight beam to be the same correct value she measured.

So, what would Bob’s spacetime diagram- his particular set of time and space axes- look like from Alice’s perspective? To start with, she’d draw her time axis as a vertical line, and her space axis as a horizontal line. She can draw her own axes however she wants, but these are the simplest choices. Now suppose Bob’s spaceship is heading to the right. When Alice draws a spacetime diagram, Bob’s position at various times traces out a line that makes an angle of 22.5˚ to the right of her time axis (because he’s traveling at half the speed of light, and half of 45 is 22.5). But when Alice draws Bob’s time and space axes, how should she draw them?

When I first tried to draw Alice’s version of Bob’s spacetime diagram, I simply drew a vertical line through Bob’s position and called that his time axis, then I drew a horizontal line through Bob’s current time and called that his space axis. Basically, all I did was redefine the zeros of Bob’s coordinate system so that he was at the intersection of the two axes.

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Then I slowly remembered that Bob defined his time axis by the line he traced out when he wasn’t moving. It also dawned on me that he’s perfectly justified to consider himself “at rest” as long as he’s sitting in his chair in the spaceship which is traveling at a constant velocity. Therefore, Bob’s time axis is actually tilted to the right of Alice’s time axis by exactly 22.5˚. From Alice’s perspective, Bob’s time axis is the same as the path traced out by his position at various times. I didn’t see any reason to alter Bob’s space axis, though, so I left it perfectly horizontal at first.

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The insight that Bob’s time axis as drawn by Alice is tilted by his velocity is a form of relativity, but it’s commonly known as Galilean relativity because Galileo published it in 1632. Unfortunately, allowing Bob’s time axis to tilt from vertical (as Alice defines her time axis) towards the 45˚ line that represents light speed implies that Bob measures light speed to be different than what Alice measures it to be. This discrepancy occurs because an observer measures an object’s speed by the angle it makes with the time and space axes. Since Alice’s and Bob’s time axes are no longer parallel, they no longer agree on the speed of light. Based on experimental results available as early as 1887 which showed that all observers must agree on the speed of light, this disagreement is a serious problem.

Einstein’s famous solution- special relativity- was to tilt Bob’s space axis “counter-clockwise” by 22.5˚, the same angle that Bob’s time axis tilted. In other words, Alice tilts Bob’s space axis towards the light speed line by the same angle that Bob’s time axis tilted towards the light speed line. By applying a tilt to Bob’s space axis that’s equal and opposite to the tilt of his time axis, Alice has insured that the line representing light speed perfectly bisects the angle formed by Bob’s time and space axes. As a result, Bob measures the speed of light to be “light speed” because the line representing light speed is midway between his time and space axes. Thanks to Einstein, Alice and Bob agree on the speed of light.

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fig4
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Unfortunately, as a result of this small change, Alice and Bob now disagree about something more fundamental. That is, they disagree about what set of events constitute “the present.” Remember that each observer’s space axis effectively defines what they consider “the present.” When Alice tilts Bob’s space axis in order to draw his spacetime diagram from her point of view, she is implicitly saying that his definition of the present is different than hers!

It may seem like Einstein solved a small problem (that Alice couldn’t draw Bob’s spacetime diagram in a way that allowed them both to agree on the speed of light) only at the cost of creating a bigger problem which violates common sense so thoroughly that Einstein simply must have been mistaken. And it’s certainly true that relativity violates common sense in other ways. The important point to remember is that special relativity has been making these extravagantly nonsensical claims for over a century, and so far every experiment performed supports special relativity’s bizarre predictions.

So how could an FTL signaling device send a message to the past? Suppose Alice and Bob both have FTL communications devices, and Bob has a spaceship with powerful sublight engines. In this scenario, Alice could stay on Earth and send a message to her own past. Here’s how it would work:

  1. Bob gets in his spaceship and travels directly away from Earth at 90% light speed for a year. The speed and travel time aren’t really important; increasing either of them just sends the message deeper into the past.
  2. Alice, on Earth, sends Bob an instantaneous message using her FTL communication device. It travels to Bob along Alice’s space axis – the line in her spacetime diagram that goes through her present position and on through Elsewhere, to define “the present.” It’s not necessary for Alice’s communication to be instantaneous, but I think it makes the argument a little clearer. Instantaneous signals might seem more outlandish than signals traveling just slightly faster than light speed, but there’s no qualitative difference according to special relativity.
  3. Bob receives the message at the exact instant (according to Alice’s timeframe) that she sent it, even though he is almost a light year away. He then immediately sends the message back to Alice using his FTL device. However, Bob is traveling at 90% of the speed of light, so his space axis is sharply tilted away from vertical- it actually intersects Alice’s “timelike past.” Bob’s “instantaneous” signal- which travels along his space axis- reaches Alice before she even sends that message to Bob in the first place.

Alice sent a message to her own past using Bob’s spaceship as a relay. Notice that this argument doesn’t refer to the particular method of FTL signaling. The FTL communications device could be literally anything- a tachyon transceiver, a traversable wormhole, a ship with an Alcubierre drive, or something more exotic. The details simply don’t matter- any FTL signaling device can be used to send messages to the past. This argument can only fail if special relativity is wrong. For instance, a preferred frame would break this argument.


Footnotes

  1. Objection: Alice and Bob were originally at rest on Earth, but Bob definitely felt the ship pushing against his back as it accelerated up to its high speed, so he must know that he’s the one who’s moving, not Alice. Alice’s diagram is therefore correct, Bob’s is wrong, and special relativity is just nonsense.
    Response 1: Why choose Earth as the official “rest frame” of the universe when there are better choices? For instance, Bob could determine Earth’s velocity (~370 km/s) relative to the CMB (loosely translated: “echo of the Big Bang”). By accelerating in the opposite direction of that velocity, he seems to have a more valid claim to being at rest, because the CMB is a relic of creation itself. But, if relativistic effects actually depended on which direction Bob accelerates, those effects should exhibit anisotropy (which hasn’t been observed).
    Response 2: Somewhat more whimsically, Bob could claim that he was only firing his engines to counteract a powerful but brief gravitational field that pulled every object in the universe in the same direction. Bob noticed this field (somehow) and accelerated in the opposite direction just to “stay at the same point.” Note that Alice wouldn’t have felt any effects because the field would affect everything at once, the same reason weightless training flights are possible.
    …anyone have a better response? ↩ back
Last modified May 8th, 2014
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10 Responses to “Any FTL signal can be sent back in time”

  1. Reythia posted on 2008-10-29 at 16:05

    Beautiful explanation of spacetime diagrams. I’d never dug into the send-yourself-a-message aspect of it, though I’ve worked on astronomy problems that are mathematically similar. Which always reminds me of why I HATED relativity in physics classes — throw “common sense” out the window before you begin, or you’re sure to fail the test!

    One of the more interesting questions I’ve had about FTL communications and travel is precisely how we would recieve such a signal or respond to it. Let’s pretend that the Vulcans decided to warp by Earth today, moving FTL but still somehow within what you and I would think of as “normal” space. (Okay, so my Star Trek analogy just broke down, but ignore that.) Using the technology we have available to us today, would we even be able to track them? I haven’t pulled out my scrap paper, but I believe the answer is yes (unless they’re moving away from us, at least). But how about about responding? How could we possibly get a message to a ship that is moving faster than any signal we know how to send?

    That’s just always bothered me. Just THINK about how many ‘Vulcans’ we may have already missed!

    • Using the technology we have available to us today, would we even be able to track them?

      It might not matter if our current technology is capable of tracking them. I say this because even if we did detect the alien ship, they could accelerate using conventional propulsion in any direction to tilt their space axis, then perform an FTL “jump” in the opposite direction to any point well outside our solar system. By doing so, they’d jump to a point in Earth’s Elsewhere region that lies before (below) the moment when they jumped away from Earth. Even though they’ve traveled back in time, they’re not quite in Earth’s timelike past yet. The reason is that the amount of time they’ve moved into the past is necessarily less than the amount of time it would take for a beam of light to reach Earth if they sent it from their current position. This means that they can’t just send a light signal back to Earth in time for it to arrive before they left Earth, thus warning themselves to avoid detection.

      However… if the aliens’ FTL jump doesn’t change their velocity in “normal space”, then upon arriving at this point in interstellar space their velocity will be pointing back towards Earth. Their next step is the same even if their velocity vector is changed somehow by the FTL jump: either way, they have to accelerate in the same direction as their original FTL jump so that their sublight velocity vector eventually points away from Earth. Now the aliens’ space axis does intersect with Earth’s past. They could then jump back to Earth- arriving before they left- and warn themselves about the upcoming detection by the savage hairless apes.

      This tactic is easiest if the aliens’ ship can perform an instantaneous jump (i.e. they can move along their space axis), but it’s also feasible even if their top speed is just a tiny bit faster than light speed. Slower FTL speeds would merely require higher sublight velocities and/or FTL travel that takes them farther away from Earth.

      This line of reasoning suggests an unsettling question. Namely, how many times have we detected alien ships, only to have them rewrite history by performing this trick?

      Having said that, I doubt that our instruments would recognize a genuine FTL signal because many types of apparent FTL signals are recognized already. The best example is a person waving a laser pointer above his head when the moon is overhead. If the beam from the laser happens to hit the moon, the beam will sweep across the moon at a speed that could easily exceed lightspeed. I wonder if the filtering done by our long-range detection instruments would just interpret low-level sensor data that implies FTL signals in a more mundane fashion- dismissing it as a “sensor ghost” or the result of something like a person waving a laser pointer around.

    • The MIT Game Lab’s open-source game A Slower Speed of Light helps players become familiar with relativistic effects, because they’re more obvious if the speed of light is slower.

    • Spacetime diagrams were finally added. Here’s the GMT script that made them.

  2. gkai posted on 2009-08-12 at 13:48

    One question I always had regarding special theory of relativity is the relation between big bang and the absence of preferred frame of reference:
    If there was a big bang, isn’t the frame attached to the center of mass of the universe “special”? Of course, this is hypothetical as it seems difficult to retrieve such a frame a posteriori….However, Doppler shifting of background microwave radiation looks like a possible candidate to, at least in theory, identify this special inertial frame, one which keep the background isotropic (or at least without dipolar non-uniformity).

    I suspect current general relativity and big-bang theory means that the center of mass of the universe has no sense, because big-bang is not really expansion in space, but expansion in space itself…Nevertheless, this is not really a satisfying explanation, is there a better vulgarisable answer to this possible come back of the aether?

    • If there was a big bang, isn’t the frame attached to the center of mass of the universe “special”? … I suspect current general relativity and big-bang theory means that the center of mass of the universe has no sense, because big-bang is not really expansion in space, but expansion in space itself…

      I also doubt such a concept is actually meaningful in general relativity, but that’s a little outside my experience (and necessary to answer this question.)

      Doppler shifting of background microwave radiation looks like a possible candidate to, at least in theory, identify this special inertial frame, one which keep the background isotropic (or at least without dipolar non-uniformity).

      Here’s a good discussion of the various components of our velocity with respect to the cosmic microwave background radiation. Assuming the primordial dipole of this radiation was negligible (which is reasonable) would allow us to calculate the exact velocity (speed and direction) in which we’d have to travel at ~370 km/s in order to be “at rest” relative to the source of the background radiation.

      But special relativity only says that physics is the same in all inertial reference frames. Some reference frames are more convenient than others, like frames fixed to the inside of the airplane when we’re aboard, the center of the Earth, the surface of the Earth, or the Sun. They’re easier to work with, but not inherently special in a way that qualifies them as preferred frames which would give relativity problems. Each frame I’ve listed simply expands the definition of “at rest” in a way that tends to appeal to our desire to generalize.

      The frame that’s at rest relative to the source of the background radiation is just the ultimate limit of that sequence. Physics would be exactly the same in that frame, even if the view was different in the microwave spectrum. I think it’s probably not special in any other sense, and certainly not a preferred frame like the aether.

      • gkai posted on 2009-08-12 at 15:25

        he he thanks for your answer!

        I agree that the notion that all physical laws have to be invariant in any inertial referential is not affected by this, so I am not sure if it has any practical implication. However, I think it is still interesting for 2 reasons:

        the historical reason: the Michelson-Morley experiment was originally devised to measure the earth velocity with respect to the aether, an easy way (or so it was believed) to measure absolute earth velocity.
        After Einstein, it was believed to be an impossible task, that the notion of absolute velocity is meaningless….

        Well, if background radiation is indeed a way to define the ultimately convenient inertial frame, then it means that the task MM tried to achieve is not impossible, the speed is 370 km/s…;-)

        The other reason is that this special reference frame (based on background radiation dipole) is theoretically measurable everywhere in the universe, and is the same (or is it?). If I’m not wrong, it means that there is a universal time, it is possible to define an absolute rate of time everywhere, regardless of local time, so unambiguous definition of simultaneity is possible (a definition that will be the same in all referentials)…Doesn’t it solve the “FTL imply time travel” paradox?

      • … the Michelson-Morley experiment was originally devised to measure the earth velocity with respect to the aether, an easy way (or so it was believed) to measure absolute earth velocity. After Einstein, it was believed to be an impossible task, that the notion of absolute velocity is meaningless … Well, if background radiation is indeed a way to define the ultimately convenient inertial frame

        The notion of absolute velocity is still meaningless. It’s just that “absolute velocity with respect to the aether” is very different from the “ultimately convenient inertial frame.”

        The other reason is that this special reference frame (based on background radiation dipole) is theoretically measurable everywhere in the universe, and is the same (or is it?).

        It’s probably meaningless on a cosmological scale, but again this is general relativity and I haven’t got enough training to say with certainty one way or the other.

        If I’m not wrong, it means that there is a universal time, it is possible to define an absolute rate of time everywhere, regardless of local time, so unambiguous definition of simultaneity is possible (a definition that will be the same in all referentials)

        Locally, it would be defined identically for observers in all galaxies, but over intergalactic distances Minkowski spacetime is inaccurate. You need to use a more advanced metric which is over my head at the moment.

        I agree that the notion that all physical laws have to be invariant in any inertial referential is not affected by this … Doesn’t it solve the “FTL imply time travel” paradox?

        I seriously doubt it; such a frame might not even be meaningful on cosmological scales. But it wouldn’t automatically define a universal time because it’s just a frame in which the (apparently) most primordial element (the CMBR) happens to be spherically symmetric. Physics wouldn’t be any different, which would be necessary to define universal time. Let me put this another way: does the frame in which the Sun is at rest define universal time inside the solar system? No; it’s more convenient to use such a frame for orbital mechanics, but that’s all. Your proposal is a lot larger in scale, but otherwise very similar.

      • gkai posted on 2009-08-13 at 05:29

        Well, it is thus that if such a convenient frame of reference exists, both Bob and Alice can use it to specify what the present is. The law of physics will still be the same in all frames, fixed speed of light will comes from the fact that it propagates in a fixed medium (the aether), and that local time and distance contraction exist based on Lorentz-Poincaré transformations for any object moving with velocity v (measured in the fixed frame or reference). This is what was proposed before Einstein (and the main equation do not change, it is mainly the interpretation that is different, it does not have the space-time 4th dimensional geometry point of view that was really the originality that Einstein introduced.) Personally I have much less trouble thinking within the aether-Lorentz framework than in the spacetime-Einstein framework….But Einstein has the advantage of getting rid of an undetectable privileged coordinate system. Now if such a detectable system can in fact be defined (background radiation)…The attraction of the old-fashioned interpretation becomes very hard to resist (at least for me, it gives my intuition (that is very good in classical physics) some rest and a way to work in problems with a high v/c…something that I am unable to do in Einsteinian interpretation…

        Do you know the document http://einsteinhoax.com/hoax.htm ? I think it present the old interpretation in a very convincing way, if you forget about the cheap attacks and a tendency to conspiracy theory ;-)

  3. oyvind posted on 2012-06-03 at 11:03

    Would it be possible to send signal back in with particles going FTL like the higgs singlet (if these particles exist) in any technolgy we know of today? I know some say this isn’t possible some other says it migth be possible (but it is long shot). And I am agree to that Einstein realtivty theory makes it possible to travel both ways back in the past and forward in the future. If it is possible to make some kind spaceship who could reach the speed of ligth or atleast 90 % it would be possible to travel in future and if it is possible to make devices who could send messages in FTL speed that would possible to send signals back in time. One paper who suggest this point of view is comming from prof. Tom Weiler and Chui Man Ho. They even suggest using the LHC in CERN. So what do you think of the those solutions? I also find David Deutsch solution regarding of the geomtry of spacetime interresting.

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