Vortex manipulator

A vortex manipulator, also colloquially called a wrist strap by Captain John Hart, (TW: Kiss Kiss, Bang Bang) was a form of basic time travel used by the Time Agency of the 51st century. It had additional capabilities as well.

Usage

Although it was generally used by the Agency, the Family of Blood stole one to follow the Tenth Doctor around the universe. (DW: Human Nature/The Family of Blood)

Time Agents were each given a vortex manipulator. Vortex manipulators used and issued by the Time Agency were housed in leather wrist-straps with a covering flap, enabling the device to be worn conveniently by the Time Agent without drawing attention to the device itself. (DW: The Empty Child)

The TARDIS controls included a vortex manipulator. (VG: Attack of the Graske)

The manipulator in the TARDIS was likely a device similar to, if less portable than the Time Agency's manipulator, performing the same function.

After Rose Tyler as Bad Wolf resurrected Jack Harkness following his extermination by the Daleks, (DW: The Parting of the Ways) Jack used his manipulator to travel from the Game Station, 200,100. He attempted to reach the 21st century Earth, but missed and ended up in 1869. The manipulator burned out after this trip. However, Jack was still able to use at least some of the other functions besides time travelling and teleporting. (DW: Utopia)

Years later, as Jack, the Tenth Doctor and Martha Jones were being attacked by Futurekind on Malcassairo in the year 100,000,000,000,000, the Doctor repaired the vortex manipulator with his sonic screwdriver, transporting them to Earth, 2008. The group used it to teleport to the Valiant, from which Martha subsequently escaped, also by use of the manipulator. (DW: The Sound of Drums) the Master later also used it to teleport from the Valiant to Earth, where he threatened to destroy the planet. The Doctor was able to use the manipulator to teleport them both back to the Valiant. Once the Master was defeated and Jack made his way back to the Torchwood Hub, the Doctor deactivated the manipulator, saying that he couldn't leave Jack with a time-travelling teleport. "You could go anywhere in time, twice; second time to apologize", according to the Doctor. (DW: Last of the Time Lords)

Captain John Hart used a manipulator to reunite with Jack and then take him to 27 A.D. Gray used his manipulator to take control of the Weevils in Cardiff. (TW: Exit Wounds)

When the Daleks relocated Earth to the Medusa Cascade and invaded it, Jack used information from the UNIT Project Indigo (based on Sontaran teleport pod technology) to reactivate the teleport function on his vortex manipulator. He then teleported to find the Doctor, arriving just after the Doctor was shot by a Dalek, which Jack then killed. (DW: The Stolen Earth) After the Daleks were defeated, the Doctor deactivated the manipulator again, saying "I told you, no teleport". (DW: Journey's End)

River Song was in possession of a Vortex Manipulator during her earlier adventures with the Doctor.

After hearing that she was in search of time travel, the salesman Dorium Maldovar met with River Song in The Maldovarium bar to offer her a Vortex manipulator that was, according to him, "Fresh off the wrist of a handsome Time Agent". She used it to teleport to Planet One at some point in the past and then to 102 A.D. on Earth. (DW: The Pandorica Opens)

After the destruction of the TARDIS, the Eleventh Doctor used River's manipulator to teleport to Easter, 1996, and then back to 102 A.D. to affect the rescue of both his earlier self and of Amy Pond. Discovering the TARDIS to be caught in a time loop at the moment of its destruction, the Doctor was able to use the manipulator to transport himself into the TARDIS to rescue River. The Doctor later time-travelled back 12 minutes to fake his death after being seriously wounded by the Stone Dalek, buying himself time to get to the Pandorica. The vortex manipulator was then used to transport the Pandorica into the heart of the explosion of the TARDIS, thus 'rebooting' the Universe. After being brought back by Amy, the Doctor gave the vortex manipulator back to River Song. (DW: The Big Bang)

When three Rivers ended up in the Doctor's TARDIS, the second one was travelling using her vortex manipulator and the Doctor sent her away before she or any of the other Rivers could see each other by activating her vortex manipulator and causing her to teleport away. (DW: Last Night)

Abilities

The main function of a vortex manipulator was to transport the user through time and space via the Time Vortex. (DW: The Empty Child, et al)

The exact range of a vortex manipulator was uncertain. One model did allow Jack Harkness to travel nearly 200,000 years (although in this instance, said manipulator burned out after doing so). (DW: Utopia) Following the Doctor's repairs and adjustments using his sonic screwdriver, the manipulator was able to travel 100 trillion years. It could transport multiple people if necessary, so long as they had some sort of physical contact with it. (DW: The Sound of Drums)

Manipulators could be seen as an extremely precise form of time travel when properly used. Both River Song and former Time Agent John Hart were able to reach their intended destinations quite regularly using vortex manipulators. When the universe was a fraction of its former size, a vortex manipulator was able to facilitate multiple precise jumps with no apparent difficulty and extreme precision - although admittedly, the distances and time differences involved were a great deal smaller. (DW: The Big Bang)

The vortex manipulator protected its user from the journey through the vortex, and was able to transport two people without difficulty or discomfort for those travelling. However transporting three people appeared to result in discomfort for the users, possibly slightly overloading the manipulator's capabilities. (DW: The Sound of Drums)

In addition to transportation, vortex manipulators had several other functions. These included:

Communication between users, including storage of messages, and listening to radio broadcasts. (TW: Kiss Kiss, Bang Bang, DW: The Sound of Drums)
Projection of holograms. (DW: The Empty Child, TW: Kiss Kiss, Bang Bang, Fragments)
Tracking of life signs. (DW: Bad Wolf, Journey's End)
Remote control of electronic devices. (TW: Everything Changes, Kiss Kiss, Bang Bang, Exit Wounds)
Muting Torchwood Three's comm system. (TW: Kiss Kiss, Bang Bang)
As a "front door key" to the Torchwood Three hub. (TW: Kiss Kiss, Bang Bang)

This seems to only apply to Jack's vortex manipulator

Playing music. (TW: Exit Wounds)
Sending a signal to enrage Weevils and to bring them back to the Cardiff sewers. (TW: Exit Wounds)
Detection of low sodium levels. (TW: Rendition)
Scanning and alteration of printed texts. (TW: Immortal Sins)
Unlocking combination-locked vaults. (TW: Immortal Sins)
Scanning a body for life, specificaly Angelo Calasanto. (TW: End of the Road)

Despite Jack's claim to Gwen Cooper that the wrist-strap was DNA coded to him, but could be reset, the fact that the Doctor, the Master and Martha were all able to use the teleporter function, as well as its detection of low sodium levels in Rex Matheson, who was simply holding the device, shows that Jack was lying in an attempt to escape. Text transmitted directly to Gwen at this time also confirmed 'Jack lies...'

Vortex manipulators were constructed from an as yet unidentified, but extremely durable metal. One was easily able to survive the detonation of a bomb which obliterated the human wearing it, albeit at the cost of the leather wrist-strap housing the vortex manipulator itself. The vortex manipulator was later recovered and installed in a new, identical wrist strap for £50 (TW: Children of Earth: Day Five) Another vortex manipulator also survived the explosion caused by Big Bang Two by being 're-booted' from its original form. However, its wearer (the Doctor) was being erased temporarily from time, which would leave the device unharmd when he returned. (DW: The Big Bang)

The Tenth Doctor considered the vortex manipulator very primitive compared to a TARDIS, describing his TARDIS as a "sports car" to Jack Harkness' "space hopper". (DW: The Sound of Drums) His eleventh incarnation described it as "cheap and nasty time travel, very bad for you", and stated that he was "trying to give it up", refencing how he's had to continuously use one to help him save the universe. (DW: The Big Bang)

Owners

Captain Jack Harkness (DW: The Empty Child)
Captain John Hart (TW: Kiss Kiss, Bang Bang)
Gray (TW: Exit Wounds)
The Family of Blood (stolen) (DW: The Family of Blood)
Mairwyn (Mexico, Time rift flotsam) (TM: Shrouded)
Dorium (from dead Time Agent) (DW: The Pandorica Opens)
River Song (acquired from Dorium Maldovar) (DW: The Pandorica Opens, The Big Bang)

Behind the scenes

Jack's Vortex Manipulator is used much in the same way as the Doctor's Sonic Screwdriver in the context of Torchwood, being capable of doing just about whatever the writers want it to. Its only known limitation is being unable to time travel or teleport, both functions having been deactivated by the Tenth Doctor.
While the Tenth Doctor forbade Jack to use the manipulator, the Ninth Doctor did not indicate any issue with it during their travels together.