# Special Relativity Questions

## Relativity and Invariance

### Question:

Are two trains, moving at different but constant velocities, in the same inertial reference frame? Do two non-inertial reference frames imply that one frame is accelerating?

"Different" velocities mean different frames. That includes the case where they have the same speed but are moving in different directions. In special relativity if a frame is noninertial it must be accelerating; it's either moving at constant velocity or it's accelerating. Two noninertial frames will also be accelerating with respect to each other, unless they are accelerating in exactly the same way (i.e. two observers could be in the same accelerating frame at different locations and at rest with respect to each other). (The situation is a bit more complicated in general relativity because inertial reference frames can generally be defined only locally.)

### Question:

Let's say that you launch two spaceships going in opposite directions. Eventually, both spaceships have a velocity of .75c, each in their respective directions. From earth, this is fine because neither speed exceeds c, but if you take the point of view of one of the spaceships, what would the velocity of the other ship be? It would seem that to one of the ships the earth would be moving away at .75c and the other ship would be moving away at 1.5c. How can this be?

Velocities don't add up like they do in Newtonian mechanics. The relativity of space and time intervals extends to velocity. Equation 7.8 describes how to add two velocities together. For your example we have (.75+.75)/(1+.75*.75) = 1.5/1.5625 = 0.96.

One can prove this relation by examining the Lorentz transform of velocity = dx/dt. This involves some mildly complicated algebra.

### Question:

OK, maybe I see why clocks should run slow, but why must there also be a length contraction?

The second postulate of relativity requires the speed of light to be the same in all reference frames. Since speed is a change of distance divided by a change of time, if time changes (time-dilation) then length must change too to maintain the constancy of light.
A question sometimes arises: is this contraction real or just an optical illusion of some sort? The answer it is real in every sense. A measurement of something moving will be shorter than of that same object at rest according to every possible test.

### Question:

Does the Concorde noticeably shrink in size, and if so how are they built to deal with that?

The shrinkage (Lorentz contraction) due to its motion is not measurable for the Concorde at the speeds it is moving. However, even if we were talking about a measurable contraction of a relativistic space ship it is not necessary to worry about Lorentz contraction in ship design. The contraction is relative, and is measured in the frame moving with respect to the ship. In the ship's own frame it has its proper length.

### Question:

Is there any way to visualize length contraction rather than just considering the mathematical equation for it?

The spacetime diagram can provide a graphical illustration of length contraction, i.e. you can see how a standard length is shorter in other frames compared to its rest frame.

### Question:

How can simultaneity be relative?

We are used to thinking of time unfolding, and time being the same for all possible observers. This isn't the way it is in SR. Simultaneity is defined as "things happening at the same time." We might also define it in this way: Two events (two points in spacetime) are simultaneous in a given frame if a light signal emitted midway between the spatial locations of those two events arrives at those events. An example would be a lightbulb flashing at the center of a train car. In the train-car's frame the light hits the front and the back of the car at the same time, hence those two events are simultaneous. But to an observer watching the train car go by at some speed the two events cannot be simultaneous because the rear of the train comes forward to meet the light signal while the front of the train moves ahead of the light signal (put another way, the flashing of the light doesn't occur midway between the two events, according to the ground-based observer, because the train is in motion).
Because events that are simultaneous must be spacelike separated, they can have no causal relationship; hence, in a sense, simultaneity is a convention. When we say that a set of clocks is synchronized we are making a statement about how the clocks are set relative to one another in a certain frame. In another frame the clocks will show different times, although all the clocks in a given inertial frame will run at the same rate.

### Question:

Why is the spacetime interval "invariant" under the Lorentz transformation? What does it mean?

The first thing to understand is that time intervals and space intervals are relative, i.e., the value an inertial observer would get for a measurement of either of them depends on the observer's frame. This is different from what we are used to thinking. Space intervals and time intervals vary between frames according to the Lorentz transformation formula (the gamma or boost factor).

If you were to take the definition for the space-time interval, plug in the lorentz transformation terms as appropriate to change delta t and delta x to a different frame, you would find that the terms cancel in such as way as to leave delta s unchanged, despite the transformation to a different frame. This means that all observers, no matter what their relative motion, will measure the same delta s even though their individual delta t and delta x intervals will be different. The agreement between all the inertial observers means that the quantity is invariant.

But what is delta s, the space time interval? In a mathematical sense, it is just a thing constructed in such a way that the Lorentz tranformation terms cancel out, that is it is designed to be an invariant quantity. Does it have a physical interpretation? It is a measure of a "distance" in spacetime. In ordinary space, a distance can be defined that is invariant under the Galilean transformation: the familiar Pythagorean rule is one such mathematical rule that gives an invariant distance even though separate observers' definition of x and y might be different. The spacetime interval extends this concept to spacetime.
There is one useful interpretation of the spacetime interval. The spacetime interval measured along the timelike worldline of an observer is equal to the proper time of that observer, i.e., what that observer's clock records. Since the spacetime interval is invariant , all observers will agree on the proper time intervals along a worldline, even though they may disagree upon the intervals of space and time along that worldline.

Since proper time is a distance in a geometry, it differs along different paths in that geometry, just as the spatial distance along a path through space depends upon the path. Consider two events in spacetime, separated by a timelike interval, and suppose that several clocks travel along separate worldlines between those two events. Each clock will measure a different proper time along its worldline, but the clock which travels along an inertial path (a straight line in spacetime) will register the longest elapsed proper time. This property of spacetime is in some respects opposite to the familiar Euclidean geometry, in which the straight line is the shortest distance between two points.
In general relativity we will invert this relationship and use the maximization of the spacetime interval to define inertial motion. Spacetimes in general relativity are curved, which means that inertial motion is not necessarily linear in the special-relativistic sense, but we can still use the maximization of proper time interval to determine which path through the spacetime represents inertial motion. But the details of this is a topic for general relativity.

### Question:

Is density relative?

Yes. Density is mass per unit volume, so if a block of some density is moving past you, length contraction would decrease the volume while the mass would increase by the relativistic equation.

### Question:

The text says that it requires too much energy to realistically reach speeds close to the speed of light. But aren't all speeds relative? Suppose I am travelling 0.7c with respect to the Earth. My ships engines and fuel are also travelling at this speed. Shouldn't it be easy to accelerate to 0.8c without too much effort? How is this more energetically expensive than accelerating from rest to 0.1c? From the ships and the engines point of view, the ship is at rest. I'm confused about the difficulties of getting closer to light speed. With respect to whom?

Speeds are indeed relative, so you must compute the change in speed from the frame of the spaceship. In this case you wish to accelerate from 0.7c to 0.8c, both measured with respect to the Earth. By the relativistic velocity addition law, the velocity change is not 0.1c in the frame of the spaceship, but is 0.23c. In the spaceship frame, the energy required is given by the corresponding boost factor, which in this case is 1.028. The additional energy needed is thus 0.028mc2. The change in boost factor as measured from the Earth frame is Gamma(.8c) = 1.67 - Gamma(.7c)=1.40; the difference is 0.27, so the additional energy required will be 0.27mc2, as measured in Earth's frame. Why the difference? The straightforward answer is that a measurement of energy is relative. However, in either frame the amount of energy required to accelerate by the same factor will increase as the ship's speed increases, because of the dependence of the boost factor upon the square of the speed. Accelerating from 0.8c to 0.9c in the Earth's frame corresponds to an increase of 0.35c in the spaceship's frame. In the spaceship frame, the energy required for the increase is 0.06mc2, while in the Earth frame it is 0.68mc2; in both cases substantially more energy is required to go from 0.8c to 0.9c than to go from 0.7c to 0.8c.

### Question:

Is there ever a case where someone's clock appears to be running faster than your own from your frame of reference?

Yes, but only if your frame of reference is an accelerated one, and/or you are deep inside a gravitational field. Gravitational time dilation then causes your clock to run slow as you see it compared to a clock up at high altitude.

### Question:

Okay if time dilation means that someone in a space ship that is moving faster than me has a slower moving clock then I do (according to the time reference of myself) then what about his point of view? If he thinks he's going at a velocity of 0 according to himself, then I should be moving at a speed greater than him and for that reason should be experiencing a slower time passage then he is. Well if this goes on for a while, who ages faster (i.e. who has the faster clock)?

It is important to realize that it really is the case that each of these people measures the other's clock as running slowly. So who is really running slowly? So long as both remain within their fixed inertial reference frames, the answer is relative and always the other guy. Ah, you say, what if they meet up again sometime? But they will never meet again if they remain in their inertial reference frames: inertial motion means linear, constant velocity travel, and straight lines meet only once. But what if they accelerate, or what if space is curved so they do meet again? In that case you can always compute how much proper time will have elapsed for each person. The result when acceleration must be considered is no longer as simple as the comparison between two inertial frames, but there need not be reciprocity between general accelerated (or curved spacetime) frames. Hence one or the other observer will be younger, and the answer is unequivocal.

### Question:

I thought that things shrinking (length contraction) was an optical illusion. But the text says that it is not. If not, then how does something shrink?

Time dilation and length contraction are not just optical illusions, but neither do they represent a physical contraction. These effects are the result of a measurement from a given inertial frame that is performed on body moving with respect to that frame. We assume that the measurements always take into account the finite travel time of light. Consider two observers moving relative to one another. You have no difficulty with the idea of their velocities being relative - each thinks the other is "really moving." In SR, time intervals and space intervals are also relative. You don't shrink or see your own clock run slow. The other observer sees your clocks slow and meter sticks contracted from his frame. Similarly you will observe his clocks slow and meters sticks short from your frame. The time dilation and length contraction are inherent properties of the way measurements must be performed in spacetime.

Note: It is a complicated question to ponder how things would appear visually if they were moving close to the speed of light. It is necessary to trace light rays carefully to determine what would actually impinge on your retina.

### Question:

If we go near the speed of light we have weird effects, time slowing, lengths contracting to zero. But light doesn't. Why isn't light subject to relativity?

Light is certainly subject to relativity. Light has zero rest mass, and relativity says that anything with zero rest mass always has to go at the speed of light along lightlike trajectories in spacetime. If you want to be anthropomorphic about it, a photon doesn't experience the passage of time. To it, it is everywhere at once.

### Question:

If a person traveling to Alpha Cen near the speed of light sees their clock running "normally" shouldn't it take him longer to reach Alpha Cen than the Earth person would believe?

No. The distance to Alpha Cen is length contracted, so even though the clock is running normally it doesn't take so long to get there.

### Question:

I'm confused about travelling to Alpha Centauri in less than 4 years on the traveller's clock. Is this due only to length contraction. Contraction of what? The "space" between Earth and Alpha Cen? What frame is that space in? Where does time dilation fit into this situation?

In the frame of the traveler, the distance between the Earth and Alpha Centrauri is length contracted. As a concrete example, suppose the spaceship is moving with a boost factor of 10 relative to the Earth. The Earth-based observer would say that the spaceship clock is running slow by a factor of 10 so it takes only 0.4 years on the spaceship clock for it to reach Alpha Cen. The spaceship, on the other hand, observes its clock to run normally but sees the Earth and Alpha Cen whipping past at a boost factor of 10, causing the distance between them to be length contracted in such an amount that it takes 0.4 years between the time the Earth passes the window until Alpha Cen appears. Both observers agree on the amount of time required for the journey in the frame of the traveler, but in one case the effect is due to time dilation and in the other, length contraction.

### Question:

All the details of special relativity, like time dilation and length contraction, are applicable in inertial reference frames. How does relativity apply in non-inertial reference frames?

Newton's Laws were formulated for inertial frames, but they work in noninertial frames so long as one adds terms to account for the accelerated frame (e.g., "fictitious" or inertial forces). The same is true for relativity. Length contraction and time dilation also occur in noninertial frames; things are just a bit more complicated. And noninertial reference frames are not equivalent to inertial reference frames, so the principle of reciprocity does not apply. (E.g. in the Twin Paradox the traveling twin is noninertial and is younger at the end of the journey; there is no symmetry between the twins.) General relativity provides the formalism to deal with all frames, including inertial, noninertial, and even curved spacetimes.

### Question:

If light doesn't move through a medium (the ether) how does it have a measurable speed?

Set up a bulb. Pace off a distance. Set up a receiver. Set up sophisticated timing apparatus. You can measure the time it takes light to travel from the bulb to the receiver. Divide by the distance. There is your speed.

### Question:

What makes the speed of light so special anyway?

Anything that has zero rest mass must travel at the speed of light. Photons carry the electromagnetic force and have zero rest mass. Now what is so special about this maximum speed? In the relativistic point of view, time and space intervals derive their meaning from interactions and physical processes which involve forces. For example, two particles interacting electromagnetically by the exchange of photon. This could happen at some maximum finite speed (which must be the same for all frames by the relativity principle), or at an infinite speed. In our universe it happens at a finite speed.

Let's speculate for a moment. Would it be possible to have a universe where interactions took place at an infinite speed? Every particle in the universe would immediately interact with every other particle. Everything would have to be fixed into some grand equilibrium. It doesn't seem to me that it would then be possible to have such a universe evolve in time.

### Question:

If the speed of light is the same in all frames, why does light travel in water at a speed less than c?

The speed of light in vacuum is the same in all frames. When light travels through water it is interacting with the water. These interactions reduce the net speed through the water.

### Question:

I have learned that the phase velocity of transverse electric waves can be greater than the speed of light in some media. How can this be? How much of scientific law is based on the idea that c is the fastest speed at which anything can travel?

The phase velocity refers to the velocity one gets by multiplying the wavelength times the frequency of a wave. It is the propagation speed of a particular wavelength component of a general wave. However, the rate at which the energy in a wave pulse propagates is the group velocity. This is also the speed at which information can be carried by a wave pulse. The group velocity is always less than c. Certain velocities can exceed c but nothing physical (information, energy etc.) can be transported faster than c.

In answer to the second part of the question, all of physical law is grounded in the principles of relativity as postulated by Einstein. It may be difficult to accept or understand, but modern technology (computers, video, internet) is based on relativity.

### Question:

Is there a relationship between F=ma and E=mc2?

One is an equation for force (Newtonian) the other for Energy (Einsteinian). They have different units: Force times distance is Energy. Even in Special Relativity you need a force to produce an acceleration. To apply a force over some distance requires energy (energy is the ability to do work). The equivalent Newtonian equation for the kinetic energy (energy of motion) is E = 1/2 m v2.

### Question:

I don't understand how a photon is energy yet not a mass even though E=mc2.

Einstein's equation as written above applies only to particles with rest mass. The photon has no rest mass, but it does have momentum and inertia and other properties normally associated with mass. Rest mass is an intrinsic property of certain particles, their "energy of being" if you will, the minimum invariant energy that they must have to exist. The photon simply requires no minimum energy to exist. In exchange, it must travel at the speed of light; a photon that stops moving ceases to exist.

### Question:

It is possible to travel to distant stars given unlimited boost factors. Assuming we had the required technology, could we survive the required acceleration?

Not that much instantaneous acceleration is required actually, just modest acceleration for a long period of time. At one gee of acceleration you could get to the Andromeda galaxy is only about 20 years as you would measure time.

### Question:

If two twins travel in opposite directions for 5 light years and then return will they be the same age?

If their journeys are symmetrical, i.e., they both go at the same speeds (relative to the Earth) and for the same time (as measured by the Earth), then yes they will be the same age (and younger compared to a third "twin" who stayed home.

### Question:

Why can't a noninertial frame be applied to special relativity?

Noninertial frames can be analyzed in special relativity; they just require more complicated equations than the ones we have presented in this book. A noninertial frame can be described in special relativity using the proper mathematical forms for relativistic acceleration. Special relativity assumes the existence of inertial frames to which these measurements are related. General relativity tells us what constitutes inertial motion in general, as well as in the presence of gravity.

### Question:

Does relativity imply that Ptolemy was right? This in the sense that a geocentric universe isn't wrong in and of itself?

Not as I see it. Ptolemaic/Aristotelian cosmology assumed the presence of a universal standard of rest, namely the Earth. Physics was different in the celestial and Earthly realms. These ideas are fundamentally counter to the relativity principle.

### Question:

In the twin Paradox: Suppose Betty takes off in one direction while Andy travels at the same speed but in the opposite direction. They travel an equal distance from Earth, then turn around and come back. What will Andy's, Betty's, and the Earth's clocks show?

Andy and Betty will have the same elapsed time (their journeys were symmetric) and they will both have less elapsed time than on the Earth.

### Question:

It is said that if you got in a spaceship and travel at a very high speed, you can get anywhere in the galaxy within your life time, although many years would have passed on Earth. Suppose you travelled in a big circle and came back to Earth. Who would have aged more seems to depend on whose perspective you take (who was really moving). What happens?

This is a variation on the twin paradox.
If you travel in a circle (at least in SR) you are accelerated, so you are not in one inertial frame the whole time. Strict reciprocity only holds between two inertial frames. When there is acceleration all observers can agree on who was "really accelerated." Both observers agree that the person who did the circular journey accelerated.
In any situation one can draw the world lines for travelers and compute the space-time interval (proper time) along their worldlines between two events (e.g. departure and reunion) and this will tell you who has aged least between those two events.

### Question:

Consider the
train in the tunnel paradox. Suppose that a bomb were rigged to go off just above the tunnel right before the train exited the tunnel, and that the train was of a fragile material that the light pulse from the bomb would destroy any part of the train not covered by the tunnel. From that perspective, it is extremely important whether or not the train ever fits entirely in the tunnel or not. Suppose that the bomb was an equal distance from each entrance. How could the light arrive at either end at a different times? What happens?

The important sentence in this thought experiment as stated, was that the blast hits both ends of the tunnel simultaneously - but this is true only in the tunnel rest frame. Remember that simultaneity is relative. In the tunnel frame the train is entirely within the tunnel when the blast hits. But an observer in the train frame sees events differently - the blast hits the front end of the tunnel before it hits the back end. Because of this different order of events, the the train still escapes destruction. But the bomb was the same distance from the front and back of the tunnel, you say. Yes, but, to the train the whole system is moving; the tunnel and the bomb are passing by at high speed. And remember that the speed of light is constant in all frames (I am assuming that it is the light pulse we are talking about here). So the situation is the same as the thought experiment in which a light bulb goes off in the center of the moving train. The pulse hits the front and back of the train simultaneously in the train frame but at two different times as measured by an observer in the ground frame.

### Question:

OK, what if the two ends of the tunnel are blocked with the train inside, but the train is made out of an infinitely tough material that cannot be compressed. Something has to give, but what?

What does it mean to be completely incompressible? It means that if you push on it, it doesn't give at all. If you had a completely incompressible rod, you could push at one end and have it move at the other end instantaneously. So is this a way to get around the finite speed of light? Make a big long rod one light year long and push on one end. If the rod cannot be compressed, then the whole thing will move at once, including the end a light year away. You have sent an instantaneous signal!
The problem is that the structural properties of something are determined by the intermolecular forces which are electromagnetic in nature. So when you push on one end of a rod, you apply forces to the molecules at that end, which in turn transmit forces on down the rod. How compressible a rod (or a train) is, is fundamentally limited by the need to transmit the force down its length, which must be limited by the speed of light. Real materials cannot evade this limit.
In the train case there is no way for the back end of the train to "know" about the front end being stopped by the blockage, except at the speed of light. If you suddenly stop the locomotive the rear of the train continues to come forward until it encounters the backward traveling signal (compression wave, shock wave, whatever). It can't avoid being crushed in this scenario.

## Space-Time Diagrams

### Question:

Why is the lightbeam on a space-time diagram placed at a 45 degree angle?

In special relativity space and time both transform between one inertial frame and another. Furthermore, they are "mixed" because the amount of length contraction or time dilation depends on a velocity, and velocity is a change in time over a change in space. It is convenient, therefore, to adopt a convention in which the units of time and space are the same, e.g. tc and x or t and x/c. Then space and time have the same footing and a light beam will correspond to a line for which delta x = c delta t, i.e. a 45 degree line. This is only for convenience, however, and has no special physical significance. The spacetime units could be meters and seconds, but in that case a light beam would be so close to horizontal on the graph that it would barely be possible to distinguish between spacelike and timelike.

### Question:

I am confused about the time and space axis of a moving observer (MO). In Figure 7.6 MO's axes do not appear perpendicular to us, but in MO's frame they are. How can this be?

Well when you think about it, what does it mean to say that the space and time axes in YOUR frame are perpendicular? Spacetime diagrams are sometimes useful for representing certain ideas, but their utility is limited by the fact that we are trying to represent graphically the properties of the geometry of Minkowski space on a flat sheet of paper. The essential point is this: all inertial reference frames are equivalent, so if we draw a spacetime diagram to illustrate our frame, we can draw another one to illustrate the moving observer's frame, and each is a valid representation of spacetime.

### Question:

What does a spacetime diagram actually mean?

It is just a way of graphically illustrating relations between points in space and time (events). In this sense it is like any other graph that might illustrate the behavior of some relationship (mathematical function).

### Question:

I understood how events on my space or time axis would be either at the same time or place, but how those events are placed on a moving observer's axes is confusing.

Think about this: if you were driving along you would think that your steering wheel was always in the same spot (right in front of you) for a long period of time. Someone watching on the side of the road says your steering wheel is not at the same spot as time goes along. So in your frame the wheel is always at the same spatial location, and in the other frame it isn't. There is nothing tricky here: the two frames are moving relative to one another. In relativity the strange thing is that something similar holds for things that are at different spatial locations at the same time in one frame (simultaneous events). In another frame these same points are at different spatial locations and different time locations.

### Question:

Pulses sent to a distant observer from near a black hole are time dilated. But the light cone is tipped toward the hole and from the distant observer's frame the light axis is nearly vertical. Doesn't that mean the pulses should come more frequently?

A vertical world line would never get out to larger radius. Those world lines tipped toward vertical take longer to get out than those directed at 45 degrees.

### Question:

How are spacetime diagrams used in cosmology and astrophysics?

In physics spacetime diagrams are very important for mapping out the interactions of particles. In cosmology we can use spacetime diagrams to determine which parts of the universe are "causally connected" - that is, given any spot in the universe, what was in that spot's past? Our backward light cone gives us the picture we have of the universe. We call this the "observable universe."

## Speculations

### Question:

Why isn't it possible to go faster than light? Why can't you just keep accelerating?

If you keep accelerating (which you can do) you go faster and faster in the sense of having greater gamma factors and more and more energy. You see the universe length-contract more and more. But you never go faster than c as you or anyone else measures it. Things interact by the exchange of massless particles. Massless particles move at the speed of light. These interactions are more fundamental than our definitions of time and space intervals. In an axiomatic sense, the second postulate of relativity means you can't go faster than light. The principle of causality would be violated if things could go faster than light. Interestingly, these things are true even in Galilean relativity, except that in Galilean relativity the speed of light is infinite.

### Question:

Have we completely and totally ruled out the possibility of faster than light travel? Is it even theoretically possible in our universe?

I have to go with "yes" on this (the ruling out part). "Going faster than light" is almost a semantic error rather than a valid physical idea. It is equivalent to asking (in Galilean relativity) "Is it possible to go faster than infinite speed?" The speed of massless particles is the ultimate speed, but instead of thinking of that as a "velocity barrier" one might do better thinking about that as a way that time and space are defined and related. The speed of light is logically equivalent to "infinite speed" as far as massive particles are concerned.

### Question:

Why is light the ultimate speed limit? If anything with mass travels less than the speed of light, what about antimatter?

The ultimate speed limit would either be some finite value, call it c, or infinity. If we adopt the relativity postulate that all inertial frames are equivalent, then the ultimate speed limit must be the same for all inertial frames. If it were infinity, then we would have Galilean relativity. If it is equal to some finite value then we have Einstein relativity. Massless particles move at the maximum speed; massive particles move at a lower speed. Since light is massless, it moves at the maximum speed.

Antimatter still has positive mass and travels slower than light. Experiments have even been done to confirm that antimatter falls in a gravitational field.

### Question:

If you go faster than light do you go back in time?

First, you can't go faster than light. Worldlines of material particles must be timelike -- this is the way that reality is constructed. The basis for the idea of traveling in time by going faster than light can be seen in the spacetime diagram. If you draw a spacelike arrow you can easily make it go back in time. You can even make it go back in time by one observer's coordinate t, and forward in time according to another observer's coordinates. Wouldn't that be strange?

### Question:

With the fictional use of light speed travel or "warp speed" in movies like Star Wars and Star Trek, what is your opinion on what it would actually be like for the people taking part in these travels. With the effects of time dilation and length contraction, I would think that everyone would get all out of sync in time and space. Any ideas on this obviously hypothetical situation?

In SciFi the "warp speed" is often superluminal. Let's leave that and concentrate instead on the more realistic idea that a space-traveling civilization could travel close to the speed of light. This is a topic treated less often by scifi writers but a very interesting one. Some examples are "The Forever War" by Joe Haldemann, and "Childhood's End" by Arthur C. Clarke.

Human lifetimes are sufficiently short that travelers would essentially leave their generation and society behind forever. What would the returning traveler experience? What if Ben Franklin had headed out on a journey and was just now returning? How many people would be willing to go on a journey knowing that not only would everyone they know be long dead when they return, but their society would probably be gone as well? This suggests that perhaps most journeys would be one-way. People would have to establish new colonies and stay there.

On the lighter side, if you left your money in an interest bearing account, when you returned you'd be pretty rich (assuming that the civilization you returned to still used money)!

### Question:

Isn't it sort of like going into the future when someone travels near the speed of light compared to the Earth? Can you experience time dilation just going around and around the Earth at high speed?

Yes, in a sense, classic Sci-Fi time travel is possible but only if you go forward. And yes you would get time dilation simply going around the Earth. (Note that this will be an accelerated frame.) We shall see (Chapter 8, 9) that the best way to get a large dilation factor is to sit close to a black hole for a while. Gravitational time dilation could quickly propel you into the Earth's distant future. What would you find then? A planet where apes evolved from men??

### Question:

Is time travel a serious idea and do physicists devote any time to studying it?

Well, yes and no. Time travel is studied as a way to investigate the implications of general relativity and possibly quantum gravity, but nobody is actually looking for a time machine. They are looking for insights into the theory under the presumption that time travel isn't possible, so if one seems to appear in a given scenario, what is wrong?

### Question:

If time travel existed what (when) would you like to visit?

Most time travel stories focus on visiting recent past events in human history (e.g., stopping John Wilkes Booth). But I would like to go back and see how the solar system formed, galaxy formed, universe formed....all your cosmological questions could be answered.

### Question:

If someday someone invents a time machine and went back in time would we notice anything different? Would we simply cease to exist?

Many sci-fi stories have been written around this idea. It seems to me that if someone did go into the past and "changed the future" then we wouldn't notice it here and now. All our memories in the altered time line would be consistent with the altered timeline and the previous timeline would cease to have ever existed. This happens all the time in Star Trek but the characters always seem to sense that "something is wrong" with the altered timeline - this seems bogus to me.

### Question:

Although traveling backward in time seems impossible, is traveling forward in time any more likely?

Well you are "traveling forward" in time right now aren't you? You could use relativistic time dilation (high speeds or strong gravity) to travel forward in time relative to (say) the Earth's frame. In that way you could time travel forward 100 years in a day. But you couldn't come back to this time to report your findings.

### Question:

Can you slow the growth of cancer by traveling close to the speed of light?

Yes, as measured by another frame, but no as measured from the point of view of the person with the cancer. All life processes proceed as "normal" according to one's proper time.

### Question:

How about a little more on tachyons, the hypothetical particle that only travels faster than light?

Let's return to Einstein's famous equation, but now write it as E2 = p2 + (moc2)2. The term p refers to the relativistic momentum. Particles with positive rest mass travel slower than light. Light has zero rest mass, and travels at the speed of light. For light E2 = p2 (light does have momentum). Now if tachyons were real, they would have rest-mass squared less than 0 because they always have v > c. This would mean that E2 - p2< 0. So if p goes up E must go down, meaning that a zero energy tachyon moves infinitely fast. You would have trouble keeping them around to observe.

If tachyons existed they could produce observable effects through interactions with matter. No such effects have ever been seen in experiments. There are also profound difficulties with the existence of tachyons in quantum field theory. The vacuum tends to break down into tachyon-antitachyon pairs. So it seems exceedingly unlikely that tachyons exist. If you are going to suggest that tachyons exist but don't interact in any way with the universe or its contents, then we are going to get into a debate about the meaning of existence. I have some pink elephants with equally valid claims.

### Question:

How does a massless body traveling at c "perceive" the universe? Would length contraction be infinite as well as time dilation and thus the body be everywhere at once?

It's rather anthropomorphic to talk of how a photon "perceives" the universe, but.... Since the spacetime interval along a lightlike trajectory is zero, there is zero proper time. The photon does not experience the passage of time (or space for that matter). The entire worldline of the photon simply is, from its beginning to its end (e.g. from the emission of the photon to its absorption).

### Question:

If it takes less time to travel somewhere the faster you go, why does it take light 4 years to go 4 lightyears?

It takes light 4 years to go 4 lightyears as you measure it. Light does it in zero proper time. (The spacetime interval along a light beam is zero.)

### Question:

If all subatomic particles in our bodies are moving at the speed of light why don't we experience the effects of time dilation?

If particles are moving at the speed of light you would need to use relativistic equations to describe their properties, but the particles in your body aren't moving near the speed of light relative to you. The subatomic scale is described by quantum mechanics rather than classical mechanics, so it isn't really accurate to think of the electron whipping around the nucleus at high speed (to take one example). The atoms in your body are pretty well described by nonrelativistic quantum mechanics. Now, as a second point, remember one would never personally experience a time dilation in the sense of thinking one's own time is running slow. One sees time dilation in another frame's clock if that frame is moving relative to yours.

### Question:

How likely is human colonization of space? Will we ever be able to travel around at high speed through the galaxy?

It's difficult to assess how likely is human colonization of space, but it seems pretty unlikely in the immediate future. The only other planet in the solar system that is remotely suitable for human habitation is Mars. Humans would not be able to live in the open on Mars, but would require spacesuits and habitable bubbles. Water would have to be extracted from permafrost. It would be a hard life even with technology, so one might ask what is the point, except for scientific exploration, or to relieve population pressures.

Colonization of extrasolar planets would be even more difficult. At the present time, we do not know of any candidates. Several generations would probably have to spend their lives on spaceships to search for and arrive at somewhat habitable planets. Where all the energy to support the space travelers would come from is not obvious; they couldn't tow a star behind them.

Energy is a major consideration of limitations on traveling at "high speed" through the galaxy. It depends, of course, on what is meant by "high speed," but presumably the expression refers to near-lightspeed travel. The amount of energy to attain such velocities, for any spaceship capable of supporting humans for extended periods (perhaps millenia), is staggering. Even at a speed of 0.99c, the distances and times involved are huge. The Galaxy is some 100,000 light years in diameter. Just to get to the center from the location of the Sun would require about 30,000 years in the Galaxy's rest frame. The travelers' time would be relativistically dilated, of course, and to them such a journey would take "only" about 4300 years, but again one might ask what is the point? There are presumably better ways to deal with population issues, though whether humanity has the self-discipline to do so is another question. Colonization could presumably preserve our species, or some descendant of it, after the Earth becomes uninhabitable due to the aging of the Sun, but ultimately the entropy of the universe itself will be too high for chemistry to occur, at which point life will become impossible anywhere. In any case, the Earth should be habitable for at least another billion years; it seems hard to imagine that any species, even an intelligent one, would last that long.

If one is still determined to undertake a program of galactic colonization, one can exchange higher speed for longer time intervals. Then the number of generations of space travelers becomes so large that one might wonder whether they could even maintain a civilization. Keep in mind that human civilization as we usually define it has existed for barely 6,000 years; we do not even know for sure that it can be maintained on Earth for tens or hundreds of thousands of years, much less on a wandering spaceship.

### Question:

What are phasers and photon torpedoes, and could they really exist?

Phasers are props for certain science fiction television shows and movies. Their method of operation is generally left vague by the writers, or else some pseudoscientific mumbo-jumbo is invoked, so whether they could exist is an open question. Photon torpedoes (in Star Trek) appear to be matter-antimatter bombs. The explosion of such a weapon would produce very high-energy photons (generally at least X-ray, but gamma ray is better) for destructive purposes.

Now as to reality: some American weapons laboratories have experimented with devices such as X-ray lasers as a kind of photon cannon. There are many technical problems with these weapons, not the least of which is obtaining and maintaining enough energy to send a sufficiently powerful beam over a great enough distance. The high energy of the photons alone is generally inadequate; there also has to be a lot of them. Another problem is that photon weapons generally dissipate most of their energy into making their path in air very hot. They were intended to be space-based weapons but this created even more acute problems with energy generation.

Returning to science fiction, the weapons would be deployed in a vacuum, removing the energy loss due to intervening molecules, but creating and transporting a large quantity of antimatter without destroying the ship would be a technological challenge, to say the least. More conventional sources of photons, such as lasers, require such a huge input of energy, only a tiny fraction of which is converted into photon energy, that their practical application in galactic weaponry is questionable.

### Question:

In "Star Wars" when the Millenium Falcon makes the "jump to lightspeed" shouldn't all those stars be blueshifted?