General Relativity question

Factual Questions
1

I am interested in physics and have a question that I’ve been unable to find the answer for. I see my question come up occasionally, but I’ve yet to see it answered. I’ve read some huge threads (both here and elsewhere) but it gets pretty old reading through hundreds of crank posts trying to find an answer.

I know that in GR there are a number of solutions that allow for closed timelike curves in certain exotic situations. I don’t care for the purpose of this OP whether such things could actually exist, but rather how the famous “grandfather paradox” arises in such a scenario. Our universe is often described as four-dimensional (3 space dimensions and 1 time dimension), but I’m beginning to suspect that this is an oversimplification. (Any insights here are welcome.) When trying to “visualize” how the grandfather paradox arises, I get hung up by the idea that if I manage to travel to an earlier time, my grandfather (quietly sitting in a rocking chair) has continued to travel forward in time as we all do under normal circumstances. In my mind, I can see myself traveling via a shortcut in spacetime to a point in the past, but I can not understand why I would find my grandfather there since he has continued to travel merrily into the future at the speed of light.

My question then, is whether some solutions of GR allow for the situation that I have described or do CTCs always give rise to causality problems? If causality issues are always a problem in these scenarios, can you explain (hopefully in English) what I am missing.

I’ve tried to keep this short, but if any further clarification is needed, just ask.

Please don’t respond with any personal “pet” theories; I’m looking for a factual answer based on existing GR theory.

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Note to mods: I see that someone else has just posted a question about the grandfather paradox. My question is a specific factual question, so I would like to keep it as a separate thread if possible.

2

As I understand your post, what you want is help mentally visualizing a certain situation that is often used to exemplify a time travel paradox–the “shooting your grandfather” bit.

What might help you is something called a “Minkowski world-diagram,” a two-dimensional representation (like the familiar Cartesian coordinate plane) in which one direction on the chart “summarizes” the three spatial dimensions of some object, and the other direction indicates that object’s extension in time. As velocity increases, the increase is reflected in a rotation of the figure representing the object; in essence, less of it occupies the space axis (so it “shrinks”), and more of it occupies the time axis (interpreted as the famous “time slowdown”). This diagrametric scheme gets one used to thinking of objects in a new way, as “lengths” in time with a comparatively small cross-section in space–everything is a time-snake. Inasmuch as one second of time equals about 186,000 miles, an average guy with a life-span of, say, 80 years would have a space-to-time ratio on the order of 1 to 80-billion, a mighty skinny snake.

This was all just preface, to get you thinking of lengths composed of a series of cross-sections. What you now think of as yourself-right-now is such a cross-section; Grandpa in the rocker (at some moment) is another.

Now look at the big picture. Gramps is a line that starts somewhere at the left of the chart, and, in the usual case, terminates (as Gramps, anyway) after a length of 80 years. You yourself are a similar line starting somewhat further to the right and running more or less parallel to the other. EXCEPT–using Goedel’s interpretation of the continuum under General Relativity, you are able to find a route into the past. On the chart, this shows up as a U-turn in your line, followed by an extension from right to left (as you head towards the past), then another U-turn as you emerge into normal time again sometime during the Roosevelt presidency. You are still running parallel to Granddad, but this new line segment, from the emergence-point onward toward the right, is next to a much earlier (ie, more to the left) segment of Granddad’s “world-line.”

Your question may be, Okay, but which point on the line is my “present”? To calculate it, you need to divide your personal line into equal segments–days, hours, seconds, whatever degree of detail you require. This is easy to figure using the ratio given above; and remember, the divisions are made along your line, not just “read off” the chart’s “grid lines.” (You don’t want 50 years of pastward journeying to seem like 50 years to you yourself, so that part of your line had better be at an angle such that one of your hours equals 50 years “normal time.”) You then find the “present present moment” and count forward–the present moment of an hour from now will be one hour further along your personal line, etc. That is the point your consciousness will occupy after you have experienced the passage of one hour. If it takes a “personal hour” to pass before you emerge into the past, that is where “the Now will be,” so to speak. And then you can shoot your Grandfather.

By relativity theory, his “Now” is (almost) exactly the point on his line that is right-angle-even with yours.

You asked about visualizing–maybe this helps. As to the resolution of the paradox, with lines apparently rewriting themselves…I’ll let others handle that.

3

Thanks Scott, that is the kind of answer that I’m looking for. I’m somewhat familiar with Minkowski diagrams and have no trouble understanding the so-called twin paradox. For some reason though, the backward time travel thing still confuses me. I think that my problem is that I am visualizing the spacetime diagram from some “outside” absolute perspective, which I know is not valid. The problem probably comes down to the fact that I can’t visualize a Lorentz transformation.

I think I followed your explanation about my worldline making a u-turn and my “present” on the worldline. Same thing as my lightcone “tipping over”, right? It just seems to me that to get to my grandfather to kill him, I need to manuever in a manner in which we are located at the same location in the spatial dimensions and have my “present” coincide with granddad’s “present”. So if I go back in time 50 years in one hour (in my frame) I have trouble understanding why granddad isn’t 50 years in my “new” future once I’m back in “normal” time. It’s as if granddad’s “present” has no physical reality and he just has an infinite number of “present times” along his worldline which may be intercepted at any point on his worldline.

It’s always that last step (there’s granddad) that seems incongruous to me. I’ve got to run for now, but will try to think about the problem in more detail later. Do you think that a Penrose diagram would help me picture the problem better?

Thanks again for your reply, it was helpful to get my brain jump started.

4

Scott’s answer is mostly correct, but it’s a bit long. To sum up, the key point is that a person doesn’t exist at only a single time. If your grandfather was born in, say, 1917, and you go back to 1938, he’ll be there. He won’t be quite the same person you know, though, in that he’ll only be 21 years old, and won’t know anything that happened after 1938. 21-year-old Gramps is in 1938, 22-year-old Gramps is in 1939, etc.

Looking at it another way, if you would expect him to “not be there anymore”, then why would you expect anything else to be there anymore, either? And if you just land in a featureless vacuum because everything’s already left, well, then, that’s not quite what I would call time travel.

5

Actually, I apreciated Scott’s long answer. I appreciate your answer as well, but if you will re-read your answer, there are a bunch of statements that are not supported by anything other than your opinion (which may be correct). What’s wrong with a “featureless vacuum” if that makes sense. It may be less interestesting than science fiction, but I’m looking for facts here.

6

heh

7

Gee, thanks… that was helpful.

8

Just to make it clear Chronos, I’m not saying that what you said is science fiction, but I would like to understand what you are saying instead of just taking it at face value. I want to understand. Is that a bad thing?

Maybe I’ve just reached the limits of my understanding without studying high level maths for a couple of years.