Many Mathematicians and physicists have made models that appear to 'make sense' and seem to hint that time travel may be possible, Frank Tipler is one of these. His design relies on the frame dragging of light cones around massive, rotating objects, specifically cylinders.

The light cone or event cone as it is sometimes known is what really gives the structure to our universe and stops everything happening at once. It is what allows us to determine a causal chain of events, and Tipler realised that extreme manipulation of light cones could allow acausal things to occur, in other words a chain of events that would need a time machine to explain them.

In regions of rotating, high gravity objects, the space in which the light cone is embedded, swirls like a whirlpool. In the same way a yacht would be tipped over by the dragging of the water, so a light cone is tipped over by the dragging of space into the direction of the rotation. If you've drawn your light cones so that their sides have a 45-degree angle, a crucial point comes when the amount of tipping reaches 45o.
Consider two events separated in space, but at the same point in time; in low gravity field areas, nothing at event A can influence the event B. Moving in toward the high gravity area around a rapidly rotating, massive object, the light cones gradually tip over, when 45o is reached, the future part of the light cone of A touches the past light cone of B. At this point a ray of light emitted from event A can travel through the past light cone of B and influence this event, even though they are actually occurring at the same time! If you increase the tipping angle, things slower than the speed of light can influence event B, which is to say a spaceship could be flown through the light cones.

To an observer in flat space-time looking in towards this high gravity whirlpool of space it would look like the orbiting spaceship was travelling faster than the speed of light. To observers in the spaceship however it seems they are only moving at their normal speed.

Frank Tipler used this concept of light cone tipping to design a time machine. His design is, take a cylinder of material, ten times the mass of the sun, and rotate it up to several billion times per second. If you then orbit close to the cylinder, with the direction of the rotation, you can travel forwards in time, and against the rotation, backwards in time.
There are several problems with this design however, firstly you need to stop the ends of the cylinder collapsing in towards each other and forming a conventional black hole. (Perhaps threading the object with matter that has a negative energy density could stop this collapse.) Secondly for the maths to make sense, apparently the cylinder should be infinitely long. And thirdly you can only ever travel backwards or forwards in time to the point of the time machines creation or destruction.

In 1974, scientist Frank Tipler published an article, "Rotating Cylinders and the Possibility of Global Causality Violation", in a reputable scientific journal, which used the theories of general relativity in a rather unique way. The article described what could be considered a "time machine" - and one that violates no known principles of physics. It used no wormholes, no black holes, no singularities.

A Tipler Cylinder is a long, (infinite, or nearly so) thin, super dense (neutron star-type material) cylinder, perhaps using about 10 times the mass of the sun. The cylinder would be spun around its axis at a speed of around a few billion revolutions per minute - at least half of the speed of light. This immense speed would be necssary to balance out the gravitational force exerted by the material, as it would be dense enough that it would otherwise collapse upon itself.

The key is that space-time is distorted by matter - the more matter in one place, the stronger the distortion. The extent of the distortion produced by this cylinder would allow something traveling at sub-light speeds to travel through time. Careful plotting of a spiral course around the cylinder will allow the time distortions to be utilized.

More detailed explanations depend on the concept of the "light-cone". A light-cone is a diagram of space-time, which represents the spatial dimensions, time, and the speed of light. They divide space-time into three areas - the past, the future, and a "forbidden" zone, delineated by the speed of light. The distortion of space-time caused by the cylinder would "tip" the light cones, eventually reaching a point where the portion representing the future would tip into the past. So as normal time progresses, it moves into the past, instead of the future, as they're overlapping.

There are some people who still believe there must be an error in the calculations somewhere, that this cannot be possible. To date, such an error has not been found. Another argument goes toward the cylinder's length - the original calculations were done with an infinite length cylinder, to simplify the calculations, but they would likely still apply to one of finite length - however, the concern is if such a finite length cylinder would collapse upon itself lengthwise due to gravity. Also, even if it is possible, then it would be limited in use - you would not be able to travel further into the past than the age of the cylinder.